https://www.math.wisc.edu/wiki/api.php?action=feedcontributions&user=Andreic&feedformat=atomUW-Math Wiki - User contributions [en]2020-07-11T11:49:52ZUser contributionsMediaWiki 1.30.1https://www.math.wisc.edu/wiki/index.php?title=Math_567_--_Elementary_Number_Theory&diff=19369Math 567 -- Elementary Number Theory2020-04-21T21:51:54Z<p>Andreic: </p>
<hr />
<div>'''MATH 567<br />
<br />
Elementary Number Theory'''<br />
<br />
MWF 1:20-2:10, Van Vleck B119<br />
<br />
'''Professor:''' [http://www.math.wisc.edu/~andreic/ Andrei Caldararu] (andreic@math.wisc.edu)<br />
<br />
''Office Hours:'' Wednesdays 2:30-3:30, Van Vleck 605. <br />
<br />
'''Grader:''' Yifan Peng, email: peng64@wisc.edu<br />
<br />
''Office Hours:'' TBA.<br />
<br />
Math 567 is a course in elementary number theory, aimed at undergraduates majoring in math or other quantitative disciplines. A general familiarity with abstract algebra at the level of Math 541 will be assumed, but students who haven't taken 541 are welcome to attend if they're willing to play a little catchup. We will be using William Stein's new (and cheap) textbook [http://www.amazon.com/Elementary-Number-Theory-Computational-Undergraduate/dp/0387855246 Elementary Number Theory: Primes, Congruences, and Secrets], which emphasizes computational approaches to the subject. If you don't need a physical copy of the book, [http://www.williamstein.org/ent/ it is available as a free legal .pdf.] We will be using the (free, public-domain) mathematical software [http://www.sagemath.org/ SAGE], developed largely by Stein, as an integral component of our coursework. There is a [http://doc.sagemath.org/pdf/en/tutorial/SageTutorial.pdf useful online tutorial.] You can download SAGE to your own computer or [http://www.sagenb.org use it online].<br />
<br />
Topics include some subset of, but are not limited to: Divisibility, the Euclidean algorithm and the GCD, linear Diophantine equations, prime numbers and uniqueness of factorization. Congruences, Chinese remainder theorem, Fermat's "little" theorem, Wilson's theorem, Euler's theorem and totient function, the RSA cryptosystem, Rabin's encryption scheme, Diffie-Hellman key exchange protocol. Number-theoretic functions, multiplicative functions, Möbius inversion. Primitive roots and indices. Quadratic reciprocity and the Legendre symbol. Perfect numbers, Mersenne primes, Fermat primes. Pythagorean triples, Special cases of Fermat's "last" theorem. Fibonacci numbers. Continued fractions. Distribution of primes, discussion of prime number theorem. Primality testing and factoring algorithms.<br />
<br />
'''Course Policies:''' Homework will be due on Fridays. It can be turned in late only with ''advance'' permission from your grader. It is acceptable to use calculators and computers on homework (indeed, some of it will require a computer) but calculators are not allowed during exams. You are encouraged to work together on homework, but writeups must be done individually.<br />
<br />
Many of the problems in this course will ask you to prove things. I expect proofs to be written in English sentences; the proofs in Stein's book are a good model for the level of verbosity I am looking for. <br />
<br />
'''Grading:''' The grade in Math 567 will be composed of 50% homework, 25% midterm, 25% final. The midterm will be on March 11, in class. The final exam will be on 5/8/2020, 7:45AM-9:45AM in room TBA.<br />
<br />
'''Syllabus:''' <br />
(This may change as we see what pace works well for the course. All section numbers refer to Stein's book.)<br />
<br />
* Jan 22-31: Prime numbers, prime factorizations, Euclidean algorithm and GCD (1.1-1.2)<br />
* Feb 3-7: The integers mod n, Euler's theorem, the phi function (2.1-2.2)<br />
* Feb 10-14: Modular exponentiation, primality testing, and primitive roots (2.4-2.5)<br />
* Feb 17-21: Public-key cryptography and RSA (3.1-3.4)<br />
* Feb 24-28: Rabin's algorithm (not in the book); algebraic numbers <br />
* Mar 2-6: Quadratic reciprocity (4.1-4.4)<br />
* Mar 9, Mar 13: Finite and infinite continued fractions (5.1-5.3)<br />
* Mar 11: ''Midterm exam''<br />
* Mar 23-30: Continued fractions and diophantine approximation (5.4-5.5)<br />
* Apr 1-3: Diophantine equations I: Pell's equation and Lagrange's theorem<br />
* Apr 6-10: More diophantine equations, elliptic curves (6.1)<br />
* Apr 13-17: Applications of elliptic curves (6.2-6.3)<br />
* Apr 20-24: More applications of elliptic curves (6.4)<br />
* Apr 27-May 1: advanced topic TBD: maybe additional discussion of cryptographic techniques?<br />
<br />
'''Video Lectures:'''<br />
Due to the university being transitioned to online teaching only, out lectures after spring break will need to be delivered online. I will post here videos for you to watch for the material we need to cover. Please watch them, and please give me feedback on how they could be improved.<br />
<br />
* '''Lecture for Mar. 23''': watch it [https://mediaspace.wisc.edu/media/Math+567+March+23+Lecture/0_fkhvkxxv here].<br />
* '''Lecture for Mar. 25''': watch it [https://mediaspace.wisc.edu/media/Math+567+March+25+lecture/0_qgkhpm19 here].<br />
* '''Lecture for Mar. 27''': watch it [https://mediaspace.wisc.edu/media/March+27+Math+567+Lecture+Video/0_ajbr2ja0 here].<br />
* '''Lecture for Mar. 30''': watch it [https://mediaspace.wisc.edu/media/March+30+Math+567+Lecture/0_yz44sw2c here].<br />
* '''Lecture for Apr. 1''': watch it [https://mediaspace.wisc.edu/media/April+1+Math+567/0_e64bdfcf here].<br />
* '''Lecture for Apr. 3''': watch it [https://mediaspace.wisc.edu/media/April+3+Math+567+Video/0_02rvfino here].<br />
* '''Lecture for Apr. 6''': watch it [https://mediaspace.wisc.edu/media/April+6+Math+567+Video/1_jmm8r268 here].<br />
* '''Lecture for Apr. 8''': watch it [https://mediaspace.wisc.edu/media/Group+law+elliptic/1_os81ndwu here].<br />
* '''Lecture for Apr. 10''': watch it [https://mediaspace.wisc.edu/media/Pollard%27s+p-1+method+for+factoring/1_qyf08tuv here].<br />
* '''Lecture for Apr. 13''': watch it [https://mediaspace.wisc.edu/media/Lenstra%2C+El+Gamal/1_848flqos here].<br />
* '''Lecture for Apr. 15''': watch it [https://mediaspace.wisc.edu/media/Arithmetic+functions/1_y6anq1l0 here].<br />
* '''Lecture for Apr. 17''': watch it [https://mediaspace.wisc.edu/media/Introduction+to+Mobius+inversion/1_e1w5jo4y here].<br />
* '''Summary of the Apr. 20 lecture''': watch it [https://mediaspace.wisc.edu/media/Summary+of+Mobius+inversion/1_hx76ks7z here].<br />
<br />
'''Homework:'''<br />
Homework is due at the beginning of class on the specified Friday. Typing your homework is not a requirement, but if you don't already know LaTeX I highly recommend that you learn it and use it to typeset your homework. I will sometimes assign extra problems, which I will e-mail to the class list and include here.<br />
<br />
* '''Jan 31''': 1.1, 1.3, 1.5, 1.7 (use SAGE), 1.8, 1.14.<br />
Problem A: Use SAGE to compute the number of x in [1..N] such that x^2 + 1 is prime, for N = 100, N = 1000, and N = 10000. Let f(N) be the number of such x. <br />
<br />
a) Can you formulate a conjecture about the relationship between f(N) and N/log N? <br />
<br />
b) What if x^2 + 1 is replaced with x^2 + 2? Can you explain why x^2 + 2 appears less likely to be prime? (Hint: consider x mod 3.)<br />
<br />
c) Prove that f(N) is at most (1/2)N+1. (Hint: consider x mod 2.) <br />
<br />
d) Give as good an upper bound as you can for f(N).<br />
<br />
Note that, despite the evident regularities you'll observe in this problem, we do not even know whether there are infinitely many primes of the form x^2 + 1! You would become very famous if you proved this.<br />
<br />
* '''Feb 7''': 2.6 (the formulation of numerical evidence should be done by Sage if you've got Sage working, and by calculator if not; you can use an online tool like [http://primes.utm.edu/curios/includes/primetest.php this] to test whether a number is prime.) 2.8,2.9,2.11,2.12,2.14,2.19<br />
<br />
* '''Feb 17''': 2.15, 2.18, 2.23, 2.26, 2.30.<br />
<br />
Problem A: Prove that if n=pq, with p,q prime, then n is not a Carmichael number.<br />
<br />
* '''Feb 21''': Book problems: 3.4, 3.5, 3.6<br />
<br />
Problem A. Prove that there are infinitely many primes p such that 2 is '''not''' a primitive root in Z/pZ. We break this up into steps.<br />
<br />
Problem A.1. Prove that, if x is an element of Z/nZ, then x^2 is not a primitive root.<br />
<br />
Problem A.2. Prove that there are infinitely many primes p such that 2 is a square in Z/pZ. Hint: suppose there are only finitely many such primes p_1, .. p_r, and define N = (p_1 .. p_r)^2 - 2. Where can you go from here...?<br />
<br />
Problem A.3. Give a list of five primes p such that 2 is not a primitive root in Z/pZ (you can use the method of this proof or any other.)<br />
<br />
Problem B. Prove that 24 is the largest integer n such that every element of (Z/nZ)^* is a root of x^2-1.<br />
<br />
* '''Feb 28''': Problem A. Using p = 23 and q=31, show how to encrypt the message x=240 with Rabin's algorithm. Find all possible decryptions of your encrypted message.<br />
<br />
* '''March 23''': 4.1 from the book.<br />
<br />
Problem A. Give a prime factorization of the Gaussian integer 7+9i.<br />
<br />
Problem B. Read the notes from [http://www.math.ucsd.edu/~jverstra/Gaussian1.pdf here]. Note that Z[i] satisfies a ''reduction theorem'': if n and d are Gaussian integers, then there exists integers q and r such that n = qd + r and Norm(r) < Norm(d). But (by contrast with the case of Z) this d may not be unique. In some contexts it is better to be able to choose r uniquely, even if this means letting r have norm greater than Norm(d). <br />
<br />
B.1. When d = 1+2i, show that, for each n in Z[i], there is a '''unique''' pair (q,r) in Z[i] such that n = qd+r and r is contained in the set {0,1,2,3,4}. For instance, i can be written as i(1+2i) + 2, so we say i reduces to 2 mod (1+2i).<br />
(Hint: Suppose q(1+2i)+r = q'(1+2i) + r'. What can we say about (r-r'), and why is this incompatible with both r and r' being in {0,1,2,3,4}?<br />
<br />
B.2. Show that if n is an integer in Z, the reduction of n mod (1+2i) is equal to its reduction mod 5.<br />
<br />
Problem C. Let's try to figure out how to define "phi(d)" for a Gaussian integer d. Suppose S is a set of Gaussian integers such that every n in Z[i] can be written uniquely as qd+r, with q in Z[i] and r in S. (So for instance when d=1+2i, we showed in problem B that we can take S to be {0...4}. It would also be OK to take S to be {1..5} or {0,i,2i,1+i,1+2i}. In fact, it turns out that S has to be a set of size Norm(d) (I might or might not prove this in class; if not, feel free just to accept it.)<br />
<br />
Now define phi(d) to be the number of elements s of S such that s and d are coprime.<br />
<br />
C.1. Compute phi(1+2i) and phi(3).<br />
<br />
C.2. Prove that the value of phi(d) does not depend on the choice of S.<br />
<br />
C.3. Prove that every n in Z[i] which is coprime to 3 satisfies n^phi(3) = 1 mod 3; that is, Euler's theorem holds in this case. (You can prove this by direct computation; of course, if you want, you are welcome to prove that Euler's theorem holds for Z[i] in general, adapting the proof in Stein's book or the one we gave in class.)<br />
<br />
*'''March 30''' <br />
<br />
Problem A. Express 50005 as the sum of two squares.<br />
<br />
In the next two problems we denote by r(n) the number of ways to express n as the sum of two squares (i.e. the number of ordered pairs (a,b) such that a^2 + b^2 = n.) For instance, r(5) = 8: (a,b) = (+/-1, +/-2) or (+/-2, +/-1).<br />
<br />
Problem B. Prove that, for any N, there exists an integer n such that r(n) > N. (I.E., the function r(n) is ''unbounded.'')<br />
<br />
Problem C. If you like Sage, write a short program in Sage to compute r(n) and compute the average of r(n) as n ranges from 1 to 1000. Whether or not you like Sage, make a guess as to how this average would behave if you replaced "1000" by a larger and larger number. (Feel free to ask the Sage-lovers what answer they got in the optional first part of the question.) Can you prove this guess is correct?<br />
<br />
Problem D. We saw in class that the ring Z[sqrt(-5)] doesn't have unique factorization; 6 can be factored as 2*3 or (1+sqrt(-5))(1-sqrt(-5)). In this problem, we will prove that Z[sqrt(-d)] fails to have unique factorization for EVERY odd d >= 5. (Actually it's true for all d >= 5 but to make the proof manageable we'll restrict to the odd case.]<br />
<br />
D.1. Show that (1+sqrt(-d)), (1-sqrt(-d)) and 2 are irreducible in Z[sqrt(-d)].<br />
<br />
D.2. Now give an element of Z[sqrt(-d)] that has two distinct factorizations into irreducibles. (Hint: imitate the example we used for Z[sqrt(-5)].)<br />
''Remark:'' The rings Z[sqrt(d)], where d is positive, are quite different -- here we believe that there are infinitely many with unique factorization, though this conjecture has remained unproved for many decades!<br />
<br />
*'''April 17'''<br />
<br />
Book problems: 5.3,5.4,5.5<br />
<br />
Problem A: We discussed in class the problem of studying which positive integers are the sum of two squares. In this problem we prove that every element n of (Z/pZ) is the sum of two squares. We argue as follows. Let S be the set of squares in (Z/pZ), and let T be the set of (Z/pZ) consisting of all elements of the form n - x^2, for some x in (Z/pZ).<br />
<br />
A.1. What is the size of S and of T? Use this to show that S and T are not disjoint.<br />
<br />
A.2. Given that S and T are not disjoint, prove that n is the sum of two squares.<br />
<br />
Problem B. Using the continued fraction expansion, find a solution to the Pell equation x^2 - 13 y^2 = 1.<br />
<br />
Problem C. Show that the modified Pell equation x^2 - 7y^2 = -1 has no solutions in integers x,y. (Hint: reduce the equation modulo a suitably chosen prime.)<br />
<br />
Problem D. Stuffy Stirnweiss finished the 1945 season with a batting average of .3085443. Using continued fractions, guess how many at-bats he had. Tony Cuccinello had a batting average of .3084577. Given that he had more than 200 and fewer than 600 at-bats, can you estimate the number of at-bats he had?<br />
<br />
*'''April 24'''<br />
<br />
Book problems 6.1, 6.2, 6.5, 6.10. <br />
<br />
Problem A.1. Pick two values of a in F_11 = Z/11Z (a not equal to 3), such that the equation y^2 = x^3+ax+1 defines an elliptic curve (i.e., it is smooth). For each such a, determine the number of points #E(F_11) and check that it falls inside the interval described in class. <br />
<br />
Problem A.2. The point P = (0,1) lies on each of these curves. For a =3 determine the order of P in the elliptic curve group, that is, find the smallest positive integer n such that nP = (infinity) -- the identity element in the group law.<br />
<br />
Problem B. Show that if A+Bi is the cube of a Gaussian integer, then A^2 + B^2 is a perfect cube.<br />
<br />
<!-- '''November 19'''<br />
<br />
Problem A. Using the method discussed in class (which is also the method of problem 4.6 in Stein, which in retrospect I think was too hard to assign with no preparation) find a nontrivial solution with y > 0 to the equation x^2 + 11y^2 = z^2. <br />
Problem B. Let f(X) be the number of solutions of A^2 + B^2 = C^3 such that A^2, B^2, and C^3 are all at most X. Use the heuristic described in class to explain why one might expect f(X) to grow more or less like X^{1/3}.<br />
<br />
<br />
Do ONE of problem D1 and D2; D1 is for people who like Sage, D2 is for people who like proving things.<br />
<br />
D1. Use Sage to compute f(X) for X = 1000, 10000, 100000. Does the answer look consistent with the heuristic prediction you made in B? Does f(X)/X^{1/3} appear to be approaching a limit?<br />
D2. We will use problem C to give a lower bound for f(X). Use this to show that there are at least X^{1/3} Gaussian integers with norm at most X that are perfect cubes. From here, show that f(X) > X^{1/3}.<br />
<br />
OPTIONAL: Show that the converse of problem C also holds -- A+Bi has norm a perfect cube if and only if A+Bi is a perfect cube. (You will need unique factorization of Gaussian integers.) Using this fact, prove that f(X) is bounded above by C X^{1/3} for some constant C.<br />
<br />
'''December 8 (note nonstandard due date)'''<br />
<br />
Problem A: Hyperbolas, ellipses, and "magic slopes"<br />
<br />
In the usual analytic geometry, both hyperbolas and ellipses are given by equations of the form<br />
<br />
ax^2 + bxy + cy^2 = d (*)<br />
<br />
<br />
(we always assume d is nonzero.)<br />
<br />
How can we tell whether such an equation describes an ellipse or a<br />
hyperbola? Well, in the geometric setting we all know and love (i.e. in<br />
R^2) a hyperbola has asymptotes and an ellipse does not. What is an <br />
asymptote? You could say it's "a line which doesn't intersect the curve", <br />
but ellipses have such lines too. You might want to say "a line which <br />
doesn't intersect the curve but comes closer and closer to the curve," but <br />
the problem here is that this a) is somewhat imprecise, and b) relies on a <br />
notion of "closer" that is not going to be very clear in Z/pZ, which is <br />
our ultimate goal!<br />
<br />
<br />
So let me state it a different way: let's say the asymptotes of a <br />
hyperbola have slopes m_1 and m_2. These are "magic slopes" in the <br />
following sense: ANY line of slope m_1 (and ditto for m_2) intersects the <br />
hyperbola in at most one point. An ellipse doesn't have "magic slopes" -- <br />
you can convince yourself by drawing pictures that, for any slope m, you <br />
can find a line of slope m striking the ellipse twice.<br />
<br />
<br />
[REMARK: For technical reasons we're going to ignore hyperbolae with a <br />
vertical asymptote, since vertical lines don't exactly have slope.]<br />
<br />
<br />
Now you might know the criterion that, over the real numbers.<br />
<br />
<br />
ax^2 + bxy + cy^2 = d<br />
<br />
<br />
is a hyperbola if b^2 - 4ac > 0 and an ellipse if b^2 - 4ac < 0. (For <br />
instance, x^2 - y^2 = 1 is a hyperbola with magic slopes 1 and -1, and x^2 <br />
+ y^2 = 1 is an ellipse.) <br />
This criterion does not make sense in Z/pZ, where there is no notion of <br />
"greater" or "less."<br />
<br />
<br />
QUESTION A.1.<br />
<br />
For which primes p is it the case that <br />
<br />
x^2 + y^2 = 1<br />
<br />
has magic slopes in Z/pZ? <br />
<br />
QUESTION A.2. For which primes p is it the case that<br />
<br />
<br />
x^2 - y^2 = 1<br />
<br />
<br />
has magic slopes in Z/pZ?<br />
<br />
<br />
QUESTION B: Back in the very first week of this course we studied the set of integers x such that x^2 + 1 is prime. Based on the heuristics we discussed in class, about how many x between 1 and N would you expect to satisfy "x^2 + 1 is prime?"<br />
<br />
<br />
QUESTION C: Let z be a complex number. Then the set of all complex numbers of the form a + bz, with a and b integers, forms a lattice in the complex plane (thought of as R^2.) What is the covolume of this lattice, in terms of z?<br />
<br />
<br />
QUESTION D: Let z be a complex number. Using Minkowski's theorem and the result of question C, show that there exist integers a,b, not both zero, such that Norm(a+bz) is at most (4/pi) Im(z). (Hint: let Omega be the set of all complex numbers of norm at most (4/pi) Im(z).)<br />
<br />
<br />
OPTIONAL EXTRA: Can you give an exact formula for the minimal norm of any nonzero complex number of the form a+bz?<br />
<br />
'''December 12''' <br />
<br />
--></div>Andreichttps://www.math.wisc.edu/wiki/index.php?title=Math_567_--_Elementary_Number_Theory&diff=19363Math 567 -- Elementary Number Theory2020-04-20T03:11:54Z<p>Andreic: </p>
<hr />
<div>'''MATH 567<br />
<br />
Elementary Number Theory'''<br />
<br />
MWF 1:20-2:10, Van Vleck B119<br />
<br />
'''Professor:''' [http://www.math.wisc.edu/~andreic/ Andrei Caldararu] (andreic@math.wisc.edu)<br />
<br />
''Office Hours:'' Wednesdays 2:30-3:30, Van Vleck 605. <br />
<br />
'''Grader:''' Yifan Peng, email: peng64@wisc.edu<br />
<br />
''Office Hours:'' TBA.<br />
<br />
Math 567 is a course in elementary number theory, aimed at undergraduates majoring in math or other quantitative disciplines. A general familiarity with abstract algebra at the level of Math 541 will be assumed, but students who haven't taken 541 are welcome to attend if they're willing to play a little catchup. We will be using William Stein's new (and cheap) textbook [http://www.amazon.com/Elementary-Number-Theory-Computational-Undergraduate/dp/0387855246 Elementary Number Theory: Primes, Congruences, and Secrets], which emphasizes computational approaches to the subject. If you don't need a physical copy of the book, [http://www.williamstein.org/ent/ it is available as a free legal .pdf.] We will be using the (free, public-domain) mathematical software [http://www.sagemath.org/ SAGE], developed largely by Stein, as an integral component of our coursework. There is a [http://doc.sagemath.org/pdf/en/tutorial/SageTutorial.pdf useful online tutorial.] You can download SAGE to your own computer or [http://www.sagenb.org use it online].<br />
<br />
Topics include some subset of, but are not limited to: Divisibility, the Euclidean algorithm and the GCD, linear Diophantine equations, prime numbers and uniqueness of factorization. Congruences, Chinese remainder theorem, Fermat's "little" theorem, Wilson's theorem, Euler's theorem and totient function, the RSA cryptosystem, Rabin's encryption scheme, Diffie-Hellman key exchange protocol. Number-theoretic functions, multiplicative functions, Möbius inversion. Primitive roots and indices. Quadratic reciprocity and the Legendre symbol. Perfect numbers, Mersenne primes, Fermat primes. Pythagorean triples, Special cases of Fermat's "last" theorem. Fibonacci numbers. Continued fractions. Distribution of primes, discussion of prime number theorem. Primality testing and factoring algorithms.<br />
<br />
'''Course Policies:''' Homework will be due on Fridays. It can be turned in late only with ''advance'' permission from your grader. It is acceptable to use calculators and computers on homework (indeed, some of it will require a computer) but calculators are not allowed during exams. You are encouraged to work together on homework, but writeups must be done individually.<br />
<br />
Many of the problems in this course will ask you to prove things. I expect proofs to be written in English sentences; the proofs in Stein's book are a good model for the level of verbosity I am looking for. <br />
<br />
'''Grading:''' The grade in Math 567 will be composed of 50% homework, 25% midterm, 25% final. The midterm will be on March 11, in class. The final exam will be on 5/8/2020, 7:45AM-9:45AM in room TBA.<br />
<br />
'''Syllabus:''' <br />
(This may change as we see what pace works well for the course. All section numbers refer to Stein's book.)<br />
<br />
* Jan 22-31: Prime numbers, prime factorizations, Euclidean algorithm and GCD (1.1-1.2)<br />
* Feb 3-7: The integers mod n, Euler's theorem, the phi function (2.1-2.2)<br />
* Feb 10-14: Modular exponentiation, primality testing, and primitive roots (2.4-2.5)<br />
* Feb 17-21: Public-key cryptography and RSA (3.1-3.4)<br />
* Feb 24-28: Rabin's algorithm (not in the book); algebraic numbers <br />
* Mar 2-6: Quadratic reciprocity (4.1-4.4)<br />
* Mar 9, Mar 13: Finite and infinite continued fractions (5.1-5.3)<br />
* Mar 11: ''Midterm exam''<br />
* Mar 23-30: Continued fractions and diophantine approximation (5.4-5.5)<br />
* Apr 1-3: Diophantine equations I: Pell's equation and Lagrange's theorem<br />
* Apr 6-10: More diophantine equations, elliptic curves (6.1)<br />
* Apr 13-17: Applications of elliptic curves (6.2-6.3)<br />
* Apr 20-24: More applications of elliptic curves (6.4)<br />
* Apr 27-May 1: advanced topic TBD: maybe additional discussion of cryptographic techniques?<br />
<br />
'''Video Lectures:'''<br />
Due to the university being transitioned to online teaching only, out lectures after spring break will need to be delivered online. I will post here videos for you to watch for the material we need to cover. Please watch them, and please give me feedback on how they could be improved.<br />
<br />
* '''Lecture for Mar. 23''': watch it [https://mediaspace.wisc.edu/media/Math+567+March+23+Lecture/0_fkhvkxxv here].<br />
* '''Lecture for Mar. 25''': watch it [https://mediaspace.wisc.edu/media/Math+567+March+25+lecture/0_qgkhpm19 here].<br />
* '''Lecture for Mar. 27''': watch it [https://mediaspace.wisc.edu/media/March+27+Math+567+Lecture+Video/0_ajbr2ja0 here].<br />
* '''Lecture for Mar. 30''': watch it [https://mediaspace.wisc.edu/media/March+30+Math+567+Lecture/0_yz44sw2c here].<br />
* '''Lecture for Apr. 1''': watch it [https://mediaspace.wisc.edu/media/April+1+Math+567/0_e64bdfcf here].<br />
* '''Lecture for Apr. 3''': watch it [https://mediaspace.wisc.edu/media/April+3+Math+567+Video/0_02rvfino here].<br />
* '''Lecture for Apr. 6''': watch it [https://mediaspace.wisc.edu/media/April+6+Math+567+Video/1_jmm8r268 here].<br />
* '''Lecture for Apr. 8''': watch it [https://mediaspace.wisc.edu/media/Group+law+elliptic/1_os81ndwu here].<br />
* '''Lecture for Apr. 10''': watch it [https://mediaspace.wisc.edu/media/Pollard%27s+p-1+method+for+factoring/1_qyf08tuv here].<br />
* '''Lecture for Apr. 13''': watch it [https://mediaspace.wisc.edu/media/Lenstra%2C+El+Gamal/1_848flqos here].<br />
* '''Lecture for Apr. 15''': watch it [https://mediaspace.wisc.edu/media/Arithmetic+functions/1_y6anq1l0 here].<br />
* '''Lecture for Apr. 17''': watch it [https://mediaspace.wisc.edu/media/Introduction+to+Mobius+inversion/1_e1w5jo4y here].<br />
<br />
'''Homework:'''<br />
Homework is due at the beginning of class on the specified Friday. Typing your homework is not a requirement, but if you don't already know LaTeX I highly recommend that you learn it and use it to typeset your homework. I will sometimes assign extra problems, which I will e-mail to the class list and include here.<br />
<br />
* '''Jan 31''': 1.1, 1.3, 1.5, 1.7 (use SAGE), 1.8, 1.14.<br />
Problem A: Use SAGE to compute the number of x in [1..N] such that x^2 + 1 is prime, for N = 100, N = 1000, and N = 10000. Let f(N) be the number of such x. <br />
<br />
a) Can you formulate a conjecture about the relationship between f(N) and N/log N? <br />
<br />
b) What if x^2 + 1 is replaced with x^2 + 2? Can you explain why x^2 + 2 appears less likely to be prime? (Hint: consider x mod 3.)<br />
<br />
c) Prove that f(N) is at most (1/2)N+1. (Hint: consider x mod 2.) <br />
<br />
d) Give as good an upper bound as you can for f(N).<br />
<br />
Note that, despite the evident regularities you'll observe in this problem, we do not even know whether there are infinitely many primes of the form x^2 + 1! You would become very famous if you proved this.<br />
<br />
* '''Feb 7''': 2.6 (the formulation of numerical evidence should be done by Sage if you've got Sage working, and by calculator if not; you can use an online tool like [http://primes.utm.edu/curios/includes/primetest.php this] to test whether a number is prime.) 2.8,2.9,2.11,2.12,2.14,2.19<br />
<br />
* '''Feb 17''': 2.15, 2.18, 2.23, 2.26, 2.30.<br />
<br />
Problem A: Prove that if n=pq, with p,q prime, then n is not a Carmichael number.<br />
<br />
* '''Feb 21''': Book problems: 3.4, 3.5, 3.6<br />
<br />
Problem A. Prove that there are infinitely many primes p such that 2 is '''not''' a primitive root in Z/pZ. We break this up into steps.<br />
<br />
Problem A.1. Prove that, if x is an element of Z/nZ, then x^2 is not a primitive root.<br />
<br />
Problem A.2. Prove that there are infinitely many primes p such that 2 is a square in Z/pZ. Hint: suppose there are only finitely many such primes p_1, .. p_r, and define N = (p_1 .. p_r)^2 - 2. Where can you go from here...?<br />
<br />
Problem A.3. Give a list of five primes p such that 2 is not a primitive root in Z/pZ (you can use the method of this proof or any other.)<br />
<br />
Problem B. Prove that 24 is the largest integer n such that every element of (Z/nZ)^* is a root of x^2-1.<br />
<br />
* '''Feb 28''': Problem A. Using p = 23 and q=31, show how to encrypt the message x=240 with Rabin's algorithm. Find all possible decryptions of your encrypted message.<br />
<br />
* '''March 23''': 4.1 from the book.<br />
<br />
Problem A. Give a prime factorization of the Gaussian integer 7+9i.<br />
<br />
Problem B. Read the notes from [http://www.math.ucsd.edu/~jverstra/Gaussian1.pdf here]. Note that Z[i] satisfies a ''reduction theorem'': if n and d are Gaussian integers, then there exists integers q and r such that n = qd + r and Norm(r) < Norm(d). But (by contrast with the case of Z) this d may not be unique. In some contexts it is better to be able to choose r uniquely, even if this means letting r have norm greater than Norm(d). <br />
<br />
B.1. When d = 1+2i, show that, for each n in Z[i], there is a '''unique''' pair (q,r) in Z[i] such that n = qd+r and r is contained in the set {0,1,2,3,4}. For instance, i can be written as i(1+2i) + 2, so we say i reduces to 2 mod (1+2i).<br />
(Hint: Suppose q(1+2i)+r = q'(1+2i) + r'. What can we say about (r-r'), and why is this incompatible with both r and r' being in {0,1,2,3,4}?<br />
<br />
B.2. Show that if n is an integer in Z, the reduction of n mod (1+2i) is equal to its reduction mod 5.<br />
<br />
Problem C. Let's try to figure out how to define "phi(d)" for a Gaussian integer d. Suppose S is a set of Gaussian integers such that every n in Z[i] can be written uniquely as qd+r, with q in Z[i] and r in S. (So for instance when d=1+2i, we showed in problem B that we can take S to be {0...4}. It would also be OK to take S to be {1..5} or {0,i,2i,1+i,1+2i}. In fact, it turns out that S has to be a set of size Norm(d) (I might or might not prove this in class; if not, feel free just to accept it.)<br />
<br />
Now define phi(d) to be the number of elements s of S such that s and d are coprime.<br />
<br />
C.1. Compute phi(1+2i) and phi(3).<br />
<br />
C.2. Prove that the value of phi(d) does not depend on the choice of S.<br />
<br />
C.3. Prove that every n in Z[i] which is coprime to 3 satisfies n^phi(3) = 1 mod 3; that is, Euler's theorem holds in this case. (You can prove this by direct computation; of course, if you want, you are welcome to prove that Euler's theorem holds for Z[i] in general, adapting the proof in Stein's book or the one we gave in class.)<br />
<br />
*'''March 30''' <br />
<br />
Problem A. Express 50005 as the sum of two squares.<br />
<br />
In the next two problems we denote by r(n) the number of ways to express n as the sum of two squares (i.e. the number of ordered pairs (a,b) such that a^2 + b^2 = n.) For instance, r(5) = 8: (a,b) = (+/-1, +/-2) or (+/-2, +/-1).<br />
<br />
Problem B. Prove that, for any N, there exists an integer n such that r(n) > N. (I.E., the function r(n) is ''unbounded.'')<br />
<br />
Problem C. If you like Sage, write a short program in Sage to compute r(n) and compute the average of r(n) as n ranges from 1 to 1000. Whether or not you like Sage, make a guess as to how this average would behave if you replaced "1000" by a larger and larger number. (Feel free to ask the Sage-lovers what answer they got in the optional first part of the question.) Can you prove this guess is correct?<br />
<br />
Problem D. We saw in class that the ring Z[sqrt(-5)] doesn't have unique factorization; 6 can be factored as 2*3 or (1+sqrt(-5))(1-sqrt(-5)). In this problem, we will prove that Z[sqrt(-d)] fails to have unique factorization for EVERY odd d >= 5. (Actually it's true for all d >= 5 but to make the proof manageable we'll restrict to the odd case.]<br />
<br />
D.1. Show that (1+sqrt(-d)), (1-sqrt(-d)) and 2 are irreducible in Z[sqrt(-d)].<br />
<br />
D.2. Now give an element of Z[sqrt(-d)] that has two distinct factorizations into irreducibles. (Hint: imitate the example we used for Z[sqrt(-5)].)<br />
''Remark:'' The rings Z[sqrt(d)], where d is positive, are quite different -- here we believe that there are infinitely many with unique factorization, though this conjecture has remained unproved for many decades!<br />
<br />
*'''April 17'''<br />
<br />
Book problems: 5.3,5.4,5.5<br />
<br />
Problem A: We discussed in class the problem of studying which positive integers are the sum of two squares. In this problem we prove that every element n of (Z/pZ) is the sum of two squares. We argue as follows. Let S be the set of squares in (Z/pZ), and let T be the set of (Z/pZ) consisting of all elements of the form n - x^2, for some x in (Z/pZ).<br />
<br />
A.1. What is the size of S and of T? Use this to show that S and T are not disjoint.<br />
<br />
A.2. Given that S and T are not disjoint, prove that n is the sum of two squares.<br />
<br />
Problem B. Using the continued fraction expansion, find a solution to the Pell equation x^2 - 13 y^2 = 1.<br />
<br />
Problem C. Show that the modified Pell equation x^2 - 7y^2 = -1 has no solutions in integers x,y. (Hint: reduce the equation modulo a suitably chosen prime.)<br />
<br />
Problem D. Stuffy Stirnweiss finished the 1945 season with a batting average of .3085443. Using continued fractions, guess how many at-bats he had. Tony Cuccinello had a batting average of .3084577. Given that he had more than 200 and fewer than 600 at-bats, can you estimate the number of at-bats he had?<br />
<br />
*'''April 24'''<br />
<br />
Book problems 6.1, 6.2, 6.5, 6.10. <br />
<br />
Problem A.1. Pick two values of a in F_11 = Z/11Z (a not equal to 3), such that the equation y^2 = x^3+ax+1 defines an elliptic curve (i.e., it is smooth). For each such a, determine the number of points #E(F_11) and check that it falls inside the interval described in class. <br />
<br />
Problem A.2. The point P = (0,1) lies on each of these curves. For a =3 determine the order of P in the elliptic curve group, that is, find the smallest positive integer n such that nP = (infinity) -- the identity element in the group law.<br />
<br />
Problem B. Show that if A+Bi is the cube of a Gaussian integer, then A^2 + B^2 is a perfect cube.<br />
<br />
<!-- '''November 19'''<br />
<br />
Problem A. Using the method discussed in class (which is also the method of problem 4.6 in Stein, which in retrospect I think was too hard to assign with no preparation) find a nontrivial solution with y > 0 to the equation x^2 + 11y^2 = z^2. <br />
Problem B. Let f(X) be the number of solutions of A^2 + B^2 = C^3 such that A^2, B^2, and C^3 are all at most X. Use the heuristic described in class to explain why one might expect f(X) to grow more or less like X^{1/3}.<br />
<br />
<br />
Do ONE of problem D1 and D2; D1 is for people who like Sage, D2 is for people who like proving things.<br />
<br />
D1. Use Sage to compute f(X) for X = 1000, 10000, 100000. Does the answer look consistent with the heuristic prediction you made in B? Does f(X)/X^{1/3} appear to be approaching a limit?<br />
D2. We will use problem C to give a lower bound for f(X). Use this to show that there are at least X^{1/3} Gaussian integers with norm at most X that are perfect cubes. From here, show that f(X) > X^{1/3}.<br />
<br />
OPTIONAL: Show that the converse of problem C also holds -- A+Bi has norm a perfect cube if and only if A+Bi is a perfect cube. (You will need unique factorization of Gaussian integers.) Using this fact, prove that f(X) is bounded above by C X^{1/3} for some constant C.<br />
<br />
'''December 8 (note nonstandard due date)'''<br />
<br />
Problem A: Hyperbolas, ellipses, and "magic slopes"<br />
<br />
In the usual analytic geometry, both hyperbolas and ellipses are given by equations of the form<br />
<br />
ax^2 + bxy + cy^2 = d (*)<br />
<br />
<br />
(we always assume d is nonzero.)<br />
<br />
How can we tell whether such an equation describes an ellipse or a<br />
hyperbola? Well, in the geometric setting we all know and love (i.e. in<br />
R^2) a hyperbola has asymptotes and an ellipse does not. What is an <br />
asymptote? You could say it's "a line which doesn't intersect the curve", <br />
but ellipses have such lines too. You might want to say "a line which <br />
doesn't intersect the curve but comes closer and closer to the curve," but <br />
the problem here is that this a) is somewhat imprecise, and b) relies on a <br />
notion of "closer" that is not going to be very clear in Z/pZ, which is <br />
our ultimate goal!<br />
<br />
<br />
So let me state it a different way: let's say the asymptotes of a <br />
hyperbola have slopes m_1 and m_2. These are "magic slopes" in the <br />
following sense: ANY line of slope m_1 (and ditto for m_2) intersects the <br />
hyperbola in at most one point. An ellipse doesn't have "magic slopes" -- <br />
you can convince yourself by drawing pictures that, for any slope m, you <br />
can find a line of slope m striking the ellipse twice.<br />
<br />
<br />
[REMARK: For technical reasons we're going to ignore hyperbolae with a <br />
vertical asymptote, since vertical lines don't exactly have slope.]<br />
<br />
<br />
Now you might know the criterion that, over the real numbers.<br />
<br />
<br />
ax^2 + bxy + cy^2 = d<br />
<br />
<br />
is a hyperbola if b^2 - 4ac > 0 and an ellipse if b^2 - 4ac < 0. (For <br />
instance, x^2 - y^2 = 1 is a hyperbola with magic slopes 1 and -1, and x^2 <br />
+ y^2 = 1 is an ellipse.) <br />
This criterion does not make sense in Z/pZ, where there is no notion of <br />
"greater" or "less."<br />
<br />
<br />
QUESTION A.1.<br />
<br />
For which primes p is it the case that <br />
<br />
x^2 + y^2 = 1<br />
<br />
has magic slopes in Z/pZ? <br />
<br />
QUESTION A.2. For which primes p is it the case that<br />
<br />
<br />
x^2 - y^2 = 1<br />
<br />
<br />
has magic slopes in Z/pZ?<br />
<br />
<br />
QUESTION B: Back in the very first week of this course we studied the set of integers x such that x^2 + 1 is prime. Based on the heuristics we discussed in class, about how many x between 1 and N would you expect to satisfy "x^2 + 1 is prime?"<br />
<br />
<br />
QUESTION C: Let z be a complex number. Then the set of all complex numbers of the form a + bz, with a and b integers, forms a lattice in the complex plane (thought of as R^2.) What is the covolume of this lattice, in terms of z?<br />
<br />
<br />
QUESTION D: Let z be a complex number. Using Minkowski's theorem and the result of question C, show that there exist integers a,b, not both zero, such that Norm(a+bz) is at most (4/pi) Im(z). (Hint: let Omega be the set of all complex numbers of norm at most (4/pi) Im(z).)<br />
<br />
<br />
OPTIONAL EXTRA: Can you give an exact formula for the minimal norm of any nonzero complex number of the form a+bz?<br />
<br />
'''December 12''' <br />
<br />
--></div>Andreichttps://www.math.wisc.edu/wiki/index.php?title=Math_567_--_Elementary_Number_Theory&diff=19362Math 567 -- Elementary Number Theory2020-04-17T17:59:50Z<p>Andreic: </p>
<hr />
<div>'''MATH 567<br />
<br />
Elementary Number Theory'''<br />
<br />
MWF 1:20-2:10, Van Vleck B119<br />
<br />
'''Professor:''' [http://www.math.wisc.edu/~andreic/ Andrei Caldararu] (andreic@math.wisc.edu)<br />
<br />
''Office Hours:'' Wednesdays 2:30-3:30, Van Vleck 605. <br />
<br />
'''Grader:''' Yifan Peng, email: peng64@wisc.edu<br />
<br />
''Office Hours:'' TBA.<br />
<br />
Math 567 is a course in elementary number theory, aimed at undergraduates majoring in math or other quantitative disciplines. A general familiarity with abstract algebra at the level of Math 541 will be assumed, but students who haven't taken 541 are welcome to attend if they're willing to play a little catchup. We will be using William Stein's new (and cheap) textbook [http://www.amazon.com/Elementary-Number-Theory-Computational-Undergraduate/dp/0387855246 Elementary Number Theory: Primes, Congruences, and Secrets], which emphasizes computational approaches to the subject. If you don't need a physical copy of the book, [http://www.williamstein.org/ent/ it is available as a free legal .pdf.] We will be using the (free, public-domain) mathematical software [http://www.sagemath.org/ SAGE], developed largely by Stein, as an integral component of our coursework. There is a [http://doc.sagemath.org/pdf/en/tutorial/SageTutorial.pdf useful online tutorial.] You can download SAGE to your own computer or [http://www.sagenb.org use it online].<br />
<br />
Topics include some subset of, but are not limited to: Divisibility, the Euclidean algorithm and the GCD, linear Diophantine equations, prime numbers and uniqueness of factorization. Congruences, Chinese remainder theorem, Fermat's "little" theorem, Wilson's theorem, Euler's theorem and totient function, the RSA cryptosystem, Rabin's encryption scheme, Diffie-Hellman key exchange protocol. Number-theoretic functions, multiplicative functions, Möbius inversion. Primitive roots and indices. Quadratic reciprocity and the Legendre symbol. Perfect numbers, Mersenne primes, Fermat primes. Pythagorean triples, Special cases of Fermat's "last" theorem. Fibonacci numbers. Continued fractions. Distribution of primes, discussion of prime number theorem. Primality testing and factoring algorithms.<br />
<br />
'''Course Policies:''' Homework will be due on Fridays. It can be turned in late only with ''advance'' permission from your grader. It is acceptable to use calculators and computers on homework (indeed, some of it will require a computer) but calculators are not allowed during exams. You are encouraged to work together on homework, but writeups must be done individually.<br />
<br />
Many of the problems in this course will ask you to prove things. I expect proofs to be written in English sentences; the proofs in Stein's book are a good model for the level of verbosity I am looking for. <br />
<br />
'''Grading:''' The grade in Math 567 will be composed of 50% homework, 25% midterm, 25% final. The midterm will be on March 11, in class. The final exam will be on 5/8/2020, 7:45AM-9:45AM in room TBA.<br />
<br />
'''Syllabus:''' <br />
(This may change as we see what pace works well for the course. All section numbers refer to Stein's book.)<br />
<br />
* Jan 22-31: Prime numbers, prime factorizations, Euclidean algorithm and GCD (1.1-1.2)<br />
* Feb 3-7: The integers mod n, Euler's theorem, the phi function (2.1-2.2)<br />
* Feb 10-14: Modular exponentiation, primality testing, and primitive roots (2.4-2.5)<br />
* Feb 17-21: Public-key cryptography and RSA (3.1-3.4)<br />
* Feb 24-28: Rabin's algorithm (not in the book); algebraic numbers <br />
* Mar 2-6: Quadratic reciprocity (4.1-4.4)<br />
* Mar 9, Mar 13: Finite and infinite continued fractions (5.1-5.3)<br />
* Mar 11: ''Midterm exam''<br />
* Mar 23-30: Continued fractions and diophantine approximation (5.4-5.5)<br />
* Apr 1-3: Diophantine equations I: Pell's equation and Lagrange's theorem<br />
* Apr 6-10: More diophantine equations, elliptic curves (6.1)<br />
* Apr 13-17: Applications of elliptic curves (6.2-6.3)<br />
* Apr 20-24: More applications of elliptic curves (6.4)<br />
* Apr 27-May 1: advanced topic TBD: maybe additional discussion of cryptographic techniques?<br />
<br />
'''Video Lectures:'''<br />
Due to the university being transitioned to online teaching only, out lectures after spring break will need to be delivered online. I will post here videos for you to watch for the material we need to cover. Please watch them, and please give me feedback on how they could be improved.<br />
<br />
* '''Lecture for Mar. 23''': watch it [https://mediaspace.wisc.edu/media/Math+567+March+23+Lecture/0_fkhvkxxv here].<br />
* '''Lecture for Mar. 25''': watch it [https://mediaspace.wisc.edu/media/Math+567+March+25+lecture/0_qgkhpm19 here].<br />
* '''Lecture for Mar. 27''': watch it [https://mediaspace.wisc.edu/media/March+27+Math+567+Lecture+Video/0_ajbr2ja0 here].<br />
* '''Lecture for Mar. 30''': watch it [https://mediaspace.wisc.edu/media/March+30+Math+567+Lecture/0_yz44sw2c here].<br />
* '''Lecture for Apr. 1''': watch it [https://mediaspace.wisc.edu/media/April+1+Math+567/0_e64bdfcf here].<br />
* '''Lecture for Apr. 3''': watch it [https://mediaspace.wisc.edu/media/April+3+Math+567+Video/0_02rvfino here].<br />
* '''Lecture for Apr. 6''': watch it [https://mediaspace.wisc.edu/media/April+6+Math+567+Video/1_jmm8r268 here].<br />
* '''Lecture for Apr. 8''': watch it [https://mediaspace.wisc.edu/media/Group+law+elliptic/1_os81ndwu here].<br />
* '''Lecture for Apr. 10''': watch it [https://mediaspace.wisc.edu/media/Pollard%27s+p-1+method+for+factoring/1_qyf08tuv here].<br />
* '''Lecture for Apr. 13''': watch it [https://mediaspace.wisc.edu/media/Lenstra%2C+El+Gamal/1_848flqos here].<br />
* '''Lecture for Apr. 15''': watch it [https://mediaspace.wisc.edu/media/Arithmetic+functions/1_y6anq1l0 here].<br />
* '''Lecture for Apr. 17''': watch it [https://mediaspace.wisc.edu/media/Introduction+to+Mobius+inversion/1_e1w5jo4y here].<br />
<br />
'''Homework:'''<br />
Homework is due at the beginning of class on the specified Friday. Typing your homework is not a requirement, but if you don't already know LaTeX I highly recommend that you learn it and use it to typeset your homework. I will sometimes assign extra problems, which I will e-mail to the class list and include here.<br />
<br />
* '''Jan 31''': 1.1, 1.3, 1.5, 1.7 (use SAGE), 1.8, 1.14.<br />
Problem A: Use SAGE to compute the number of x in [1..N] such that x^2 + 1 is prime, for N = 100, N = 1000, and N = 10000. Let f(N) be the number of such x. <br />
<br />
a) Can you formulate a conjecture about the relationship between f(N) and N/log N? <br />
<br />
b) What if x^2 + 1 is replaced with x^2 + 2? Can you explain why x^2 + 2 appears less likely to be prime? (Hint: consider x mod 3.)<br />
<br />
c) Prove that f(N) is at most (1/2)N+1. (Hint: consider x mod 2.) <br />
<br />
d) Give as good an upper bound as you can for f(N).<br />
<br />
Note that, despite the evident regularities you'll observe in this problem, we do not even know whether there are infinitely many primes of the form x^2 + 1! You would become very famous if you proved this.<br />
<br />
* '''Feb 7''': 2.6 (the formulation of numerical evidence should be done by Sage if you've got Sage working, and by calculator if not; you can use an online tool like [http://primes.utm.edu/curios/includes/primetest.php this] to test whether a number is prime.) 2.8,2.9,2.11,2.12,2.14,2.19<br />
<br />
* '''Feb 17''': 2.15, 2.18, 2.23, 2.26, 2.30.<br />
<br />
Problem A: Prove that if n=pq, with p,q prime, then n is not a Carmichael number.<br />
<br />
* '''Feb 21''': Book problems: 3.4, 3.5, 3.6<br />
<br />
Problem A. Prove that there are infinitely many primes p such that 2 is '''not''' a primitive root in Z/pZ. We break this up into steps.<br />
<br />
Problem A.1. Prove that, if x is an element of Z/nZ, then x^2 is not a primitive root.<br />
<br />
Problem A.2. Prove that there are infinitely many primes p such that 2 is a square in Z/pZ. Hint: suppose there are only finitely many such primes p_1, .. p_r, and define N = (p_1 .. p_r)^2 - 2. Where can you go from here...?<br />
<br />
Problem A.3. Give a list of five primes p such that 2 is not a primitive root in Z/pZ (you can use the method of this proof or any other.)<br />
<br />
Problem B. Prove that 24 is the largest integer n such that every element of (Z/nZ)^* is a root of x^2-1.<br />
<br />
* '''Feb 28''': Problem A. Using p = 23 and q=31, show how to encrypt the message x=240 with Rabin's algorithm. Find all possible decryptions of your encrypted message.<br />
<br />
* '''March 23''': 4.1 from the book.<br />
<br />
Problem A. Give a prime factorization of the Gaussian integer 7+9i.<br />
<br />
Problem B. Read the notes from [http://www.math.ucsd.edu/~jverstra/Gaussian1.pdf here]. Note that Z[i] satisfies a ''reduction theorem'': if n and d are Gaussian integers, then there exists integers q and r such that n = qd + r and Norm(r) < Norm(d). But (by contrast with the case of Z) this d may not be unique. In some contexts it is better to be able to choose r uniquely, even if this means letting r have norm greater than Norm(d). <br />
<br />
B.1. When d = 1+2i, show that, for each n in Z[i], there is a '''unique''' pair (q,r) in Z[i] such that n = qd+r and r is contained in the set {0,1,2,3,4}. For instance, i can be written as i(1+2i) + 2, so we say i reduces to 2 mod (1+2i).<br />
(Hint: Suppose q(1+2i)+r = q'(1+2i) + r'. What can we say about (r-r'), and why is this incompatible with both r and r' being in {0,1,2,3,4}?<br />
<br />
B.2. Show that if n is an integer in Z, the reduction of n mod (1+2i) is equal to its reduction mod 5.<br />
<br />
Problem C. Let's try to figure out how to define "phi(d)" for a Gaussian integer d. Suppose S is a set of Gaussian integers such that every n in Z[i] can be written uniquely as qd+r, with q in Z[i] and r in S. (So for instance when d=1+2i, we showed in problem B that we can take S to be {0...4}. It would also be OK to take S to be {1..5} or {0,i,2i,1+i,1+2i}. In fact, it turns out that S has to be a set of size Norm(d) (I might or might not prove this in class; if not, feel free just to accept it.)<br />
<br />
Now define phi(d) to be the number of elements s of S such that s and d are coprime.<br />
<br />
C.1. Compute phi(1+2i) and phi(3).<br />
<br />
C.2. Prove that the value of phi(d) does not depend on the choice of S.<br />
<br />
C.3. Prove that every n in Z[i] which is coprime to 3 satisfies n^phi(3) = 1 mod 3; that is, Euler's theorem holds in this case. (You can prove this by direct computation; of course, if you want, you are welcome to prove that Euler's theorem holds for Z[i] in general, adapting the proof in Stein's book or the one we gave in class.)<br />
<br />
*'''March 30''' <br />
<br />
Problem A. Express 50005 as the sum of two squares.<br />
<br />
In the next two problems we denote by r(n) the number of ways to express n as the sum of two squares (i.e. the number of ordered pairs (a,b) such that a^2 + b^2 = n.) For instance, r(5) = 8: (a,b) = (+/-1, +/-2) or (+/-2, +/-1).<br />
<br />
Problem B. Prove that, for any N, there exists an integer n such that r(n) > N. (I.E., the function r(n) is ''unbounded.'')<br />
<br />
Problem C. If you like Sage, write a short program in Sage to compute r(n) and compute the average of r(n) as n ranges from 1 to 1000. Whether or not you like Sage, make a guess as to how this average would behave if you replaced "1000" by a larger and larger number. (Feel free to ask the Sage-lovers what answer they got in the optional first part of the question.) Can you prove this guess is correct?<br />
<br />
Problem D. We saw in class that the ring Z[sqrt(-5)] doesn't have unique factorization; 6 can be factored as 2*3 or (1+sqrt(-5))(1-sqrt(-5)). In this problem, we will prove that Z[sqrt(-d)] fails to have unique factorization for EVERY odd d >= 5. (Actually it's true for all d >= 5 but to make the proof manageable we'll restrict to the odd case.]<br />
<br />
D.1. Show that (1+sqrt(-d)), (1-sqrt(-d)) and 2 are irreducible in Z[sqrt(-d)].<br />
<br />
D.2. Now give an element of Z[sqrt(-d)] that has two distinct factorizations into irreducibles. (Hint: imitate the example we used for Z[sqrt(-5)].)<br />
''Remark:'' The rings Z[sqrt(d)], where d is positive, are quite different -- here we believe that there are infinitely many with unique factorization, though this conjecture has remained unproved for many decades!<br />
<br />
*'''April 17'''<br />
<br />
Book problems: 5.3,5.4,5.5<br />
<br />
Problem A: We discussed in class the problem of studying which positive integers are the sum of two squares. In this problem we prove that every element n of (Z/pZ) is the sum of two squares. We argue as follows. Let S be the set of squares in (Z/pZ), and let T be the set of (Z/pZ) consisting of all elements of the form n - x^2, for some x in (Z/pZ).<br />
<br />
A.1. What is the size of S and of T? Use this to show that S and T are not disjoint.<br />
<br />
A.2. Given that S and T are not disjoint, prove that n is the sum of two squares.<br />
<br />
Problem B. Using the continued fraction expansion, find a solution to the Pell equation x^2 - 13 y^2 = 1.<br />
<br />
Problem C. Show that the modified Pell equation x^2 - 7y^2 = -1 has no solutions in integers x,y. (Hint: reduce the equation modulo a suitably chosen prime.)<br />
<br />
Problem D. Stuffy Stirnweiss finished the 1945 season with a batting average of .3085443. Using continued fractions, guess how many at-bats he had. Tony Cuccinello had a batting average of .3084577. Given that he had more than 200 and fewer than 600 at-bats, can you estimate the number of at-bats he had?<br />
<br />
<!--<br />
'''December 7'''<br />
<br />
Problem A. When p is a prime congruent to 3 mod 4, prove that ((p-1)/2)! is either 1 or -1 in (Z/pZ). '''OPTIONAL:''' Use sage to compute whether this factorial is 1 or -1 for many primes p. Is there any pattern? Does it seem to be 1 half the time and -1 half the time? Note that I have no idea what the answer to this question is.<br />
<br />
Problem B. Using the continued fraction expansion, find a solution to the Pell equation x^2 - 13 y^2 = 1.<br />
<br />
Problem C. Show that the modified Pell equation x^2 - 7y^2 = -1 has no solutions in integers x,y. (Hint: reduce the equation modulo a suitably chosen prime.)<br />
<br />
Problem D.1. Pick two values of a in F_11 = Z/11Z (a not equal to 3), such that the equation y^2 = x^3+ax+1 defines an elliptic curve (i.e., it is smooth). For each such a, determine the number of points #E(F_11) and check that it falls inside the interval described in class. <br />
<br />
Problem D.2. The point P = (0,1) lies on each of these curves. For a =3 (the curve discussed in class) determine the order of P in the elliptic curve group, that is, find the smallest positive integer n such that nP = (infinity) -- the identity element in the group law.<br />
<br />
<!-- '''November 19'''<br />
<br />
Problem A. Using the method discussed in class (which is also the method of problem 4.6 in Stein, which in retrospect I think was too hard to assign with no preparation) find a nontrivial solution with y > 0 to the equation x^2 + 11y^2 = z^2. <br />
Problem B. Let f(X) be the number of solutions of A^2 + B^2 = C^3 such that A^2, B^2, and C^3 are all at most X. Use the heuristic described in class to explain why one might expect f(X) to grow more or less like X^{1/3}.<br />
<br />
Problem C. Show that if A+Bi is the cube of a Gaussian integer, then A^2 + B^2 is a perfect cube.<br />
<br />
Do ONE of problem D1 and D2; D1 is for people who like Sage, D2 is for people who like proving things.<br />
<br />
D1. Use Sage to compute f(X) for X = 1000, 10000, 100000. Does the answer look consistent with the heuristic prediction you made in B? Does f(X)/X^{1/3} appear to be approaching a limit?<br />
D2. We will use problem C to give a lower bound for f(X). Use this to show that there are at least X^{1/3} Gaussian integers with norm at most X that are perfect cubes. From here, show that f(X) > X^{1/3}.<br />
<br />
OPTIONAL: Show that the converse of problem C also holds -- A+Bi has norm a perfect cube if and only if A+Bi is a perfect cube. (You will need unique factorization of Gaussian integers.) Using this fact, prove that f(X) is bounded above by C X^{1/3} for some constant C.<br />
<br />
'''December 8 (note nonstandard due date)'''<br />
<br />
Problem A: Hyperbolas, ellipses, and "magic slopes"<br />
<br />
In the usual analytic geometry, both hyperbolas and ellipses are given by equations of the form<br />
<br />
ax^2 + bxy + cy^2 = d (*)<br />
<br />
<br />
(we always assume d is nonzero.)<br />
<br />
How can we tell whether such an equation describes an ellipse or a<br />
hyperbola? Well, in the geometric setting we all know and love (i.e. in<br />
R^2) a hyperbola has asymptotes and an ellipse does not. What is an <br />
asymptote? You could say it's "a line which doesn't intersect the curve", <br />
but ellipses have such lines too. You might want to say "a line which <br />
doesn't intersect the curve but comes closer and closer to the curve," but <br />
the problem here is that this a) is somewhat imprecise, and b) relies on a <br />
notion of "closer" that is not going to be very clear in Z/pZ, which is <br />
our ultimate goal!<br />
<br />
<br />
So let me state it a different way: let's say the asymptotes of a <br />
hyperbola have slopes m_1 and m_2. These are "magic slopes" in the <br />
following sense: ANY line of slope m_1 (and ditto for m_2) intersects the <br />
hyperbola in at most one point. An ellipse doesn't have "magic slopes" -- <br />
you can convince yourself by drawing pictures that, for any slope m, you <br />
can find a line of slope m striking the ellipse twice.<br />
<br />
<br />
[REMARK: For technical reasons we're going to ignore hyperbolae with a <br />
vertical asymptote, since vertical lines don't exactly have slope.]<br />
<br />
<br />
Now you might know the criterion that, over the real numbers.<br />
<br />
<br />
ax^2 + bxy + cy^2 = d<br />
<br />
<br />
is a hyperbola if b^2 - 4ac > 0 and an ellipse if b^2 - 4ac < 0. (For <br />
instance, x^2 - y^2 = 1 is a hyperbola with magic slopes 1 and -1, and x^2 <br />
+ y^2 = 1 is an ellipse.) <br />
This criterion does not make sense in Z/pZ, where there is no notion of <br />
"greater" or "less."<br />
<br />
<br />
QUESTION A.1.<br />
<br />
For which primes p is it the case that <br />
<br />
x^2 + y^2 = 1<br />
<br />
has magic slopes in Z/pZ? <br />
<br />
QUESTION A.2. For which primes p is it the case that<br />
<br />
<br />
x^2 - y^2 = 1<br />
<br />
<br />
has magic slopes in Z/pZ?<br />
<br />
<br />
QUESTION B: Back in the very first week of this course we studied the set of integers x such that x^2 + 1 is prime. Based on the heuristics we discussed in class, about how many x between 1 and N would you expect to satisfy "x^2 + 1 is prime?"<br />
<br />
<br />
QUESTION C: Let z be a complex number. Then the set of all complex numbers of the form a + bz, with a and b integers, forms a lattice in the complex plane (thought of as R^2.) What is the covolume of this lattice, in terms of z?<br />
<br />
<br />
QUESTION D: Let z be a complex number. Using Minkowski's theorem and the result of question C, show that there exist integers a,b, not both zero, such that Norm(a+bz) is at most (4/pi) Im(z). (Hint: let Omega be the set of all complex numbers of norm at most (4/pi) Im(z).)<br />
<br />
<br />
OPTIONAL EXTRA: Can you give an exact formula for the minimal norm of any nonzero complex number of the form a+bz?<br />
<br />
'''December 12''' Book problems 6.1, 6.2, 6.5, 6.10. For extra credit attempt to do problem 6.9.<br />
<br />
--></div>Andreichttps://www.math.wisc.edu/wiki/index.php?title=Math_567_--_Elementary_Number_Theory&diff=19356Math 567 -- Elementary Number Theory2020-04-15T05:02:37Z<p>Andreic: </p>
<hr />
<div>'''MATH 567<br />
<br />
Elementary Number Theory'''<br />
<br />
MWF 1:20-2:10, Van Vleck B119<br />
<br />
'''Professor:''' [http://www.math.wisc.edu/~andreic/ Andrei Caldararu] (andreic@math.wisc.edu)<br />
<br />
''Office Hours:'' Wednesdays 2:30-3:30, Van Vleck 605. <br />
<br />
'''Grader:''' Yifan Peng, email: peng64@wisc.edu<br />
<br />
''Office Hours:'' TBA.<br />
<br />
Math 567 is a course in elementary number theory, aimed at undergraduates majoring in math or other quantitative disciplines. A general familiarity with abstract algebra at the level of Math 541 will be assumed, but students who haven't taken 541 are welcome to attend if they're willing to play a little catchup. We will be using William Stein's new (and cheap) textbook [http://www.amazon.com/Elementary-Number-Theory-Computational-Undergraduate/dp/0387855246 Elementary Number Theory: Primes, Congruences, and Secrets], which emphasizes computational approaches to the subject. If you don't need a physical copy of the book, [http://www.williamstein.org/ent/ it is available as a free legal .pdf.] We will be using the (free, public-domain) mathematical software [http://www.sagemath.org/ SAGE], developed largely by Stein, as an integral component of our coursework. There is a [http://doc.sagemath.org/pdf/en/tutorial/SageTutorial.pdf useful online tutorial.] You can download SAGE to your own computer or [http://www.sagenb.org use it online].<br />
<br />
Topics include some subset of, but are not limited to: Divisibility, the Euclidean algorithm and the GCD, linear Diophantine equations, prime numbers and uniqueness of factorization. Congruences, Chinese remainder theorem, Fermat's "little" theorem, Wilson's theorem, Euler's theorem and totient function, the RSA cryptosystem, Rabin's encryption scheme, Diffie-Hellman key exchange protocol. Number-theoretic functions, multiplicative functions, Möbius inversion. Primitive roots and indices. Quadratic reciprocity and the Legendre symbol. Perfect numbers, Mersenne primes, Fermat primes. Pythagorean triples, Special cases of Fermat's "last" theorem. Fibonacci numbers. Continued fractions. Distribution of primes, discussion of prime number theorem. Primality testing and factoring algorithms.<br />
<br />
'''Course Policies:''' Homework will be due on Fridays. It can be turned in late only with ''advance'' permission from your grader. It is acceptable to use calculators and computers on homework (indeed, some of it will require a computer) but calculators are not allowed during exams. You are encouraged to work together on homework, but writeups must be done individually.<br />
<br />
Many of the problems in this course will ask you to prove things. I expect proofs to be written in English sentences; the proofs in Stein's book are a good model for the level of verbosity I am looking for. <br />
<br />
'''Grading:''' The grade in Math 567 will be composed of 50% homework, 25% midterm, 25% final. The midterm will be on March 11, in class. The final exam will be on 5/8/2020, 7:45AM-9:45AM in room TBA.<br />
<br />
'''Syllabus:''' <br />
(This may change as we see what pace works well for the course. All section numbers refer to Stein's book.)<br />
<br />
* Jan 22-31: Prime numbers, prime factorizations, Euclidean algorithm and GCD (1.1-1.2)<br />
* Feb 3-7: The integers mod n, Euler's theorem, the phi function (2.1-2.2)<br />
* Feb 10-14: Modular exponentiation, primality testing, and primitive roots (2.4-2.5)<br />
* Feb 17-21: Public-key cryptography and RSA (3.1-3.4)<br />
* Feb 24-28: Rabin's algorithm (not in the book); algebraic numbers <br />
* Mar 2-6: Quadratic reciprocity (4.1-4.4)<br />
* Mar 9, Mar 13: Finite and infinite continued fractions (5.1-5.3)<br />
* Mar 11: ''Midterm exam''<br />
* Mar 23-30: Continued fractions and diophantine approximation (5.4-5.5)<br />
* Apr 1-3: Diophantine equations I: Pell's equation and Lagrange's theorem<br />
* Apr 6-10: More diophantine equations, elliptic curves (6.1)<br />
* Apr 13-17: Applications of elliptic curves (6.2-6.3)<br />
* Apr 20-24: More applications of elliptic curves (6.4)<br />
* Apr 27-May 1: advanced topic TBD: maybe additional discussion of cryptographic techniques?<br />
<br />
'''Video Lectures:'''<br />
Due to the university being transitioned to online teaching only, out lectures after spring break will need to be delivered online. I will post here videos for you to watch for the material we need to cover. Please watch them, and please give me feedback on how they could be improved.<br />
<br />
* '''Lecture for Mar. 23''': watch it [https://mediaspace.wisc.edu/media/Math+567+March+23+Lecture/0_fkhvkxxv here].<br />
* '''Lecture for Mar. 25''': watch it [https://mediaspace.wisc.edu/media/Math+567+March+25+lecture/0_qgkhpm19 here].<br />
* '''Lecture for Mar. 27''': watch it [https://mediaspace.wisc.edu/media/March+27+Math+567+Lecture+Video/0_ajbr2ja0 here].<br />
* '''Lecture for Mar. 30''': watch it [https://mediaspace.wisc.edu/media/March+30+Math+567+Lecture/0_yz44sw2c here].<br />
* '''Lecture for Apr. 1''': watch it [https://mediaspace.wisc.edu/media/April+1+Math+567/0_e64bdfcf here].<br />
* '''Lecture for Apr. 3''': watch it [https://mediaspace.wisc.edu/media/April+3+Math+567+Video/0_02rvfino here].<br />
* '''Lecture for Apr. 6''': watch it [https://mediaspace.wisc.edu/media/April+6+Math+567+Video/1_jmm8r268 here].<br />
* '''Lecture for Apr. 8''': watch it [https://mediaspace.wisc.edu/media/Group+law+elliptic/1_os81ndwu here].<br />
* '''Lecture for Apr. 10''': watch it [https://mediaspace.wisc.edu/media/Pollard%27s+p-1+method+for+factoring/1_qyf08tuv here].<br />
* '''Lecture for Apr. 13''': watch it [https://mediaspace.wisc.edu/media/Lenstra%2C+El+Gamal/1_848flqos here].<br />
* '''Lecture for Apr. 15''': watch it [https://mediaspace.wisc.edu/media/Arithmetic+functions/1_y6anq1l0 here].<br />
<br />
'''Homework:'''<br />
Homework is due at the beginning of class on the specified Friday. Typing your homework is not a requirement, but if you don't already know LaTeX I highly recommend that you learn it and use it to typeset your homework. I will sometimes assign extra problems, which I will e-mail to the class list and include here.<br />
<br />
* '''Jan 31''': 1.1, 1.3, 1.5, 1.7 (use SAGE), 1.8, 1.14.<br />
Problem A: Use SAGE to compute the number of x in [1..N] such that x^2 + 1 is prime, for N = 100, N = 1000, and N = 10000. Let f(N) be the number of such x. <br />
<br />
a) Can you formulate a conjecture about the relationship between f(N) and N/log N? <br />
<br />
b) What if x^2 + 1 is replaced with x^2 + 2? Can you explain why x^2 + 2 appears less likely to be prime? (Hint: consider x mod 3.)<br />
<br />
c) Prove that f(N) is at most (1/2)N+1. (Hint: consider x mod 2.) <br />
<br />
d) Give as good an upper bound as you can for f(N).<br />
<br />
Note that, despite the evident regularities you'll observe in this problem, we do not even know whether there are infinitely many primes of the form x^2 + 1! You would become very famous if you proved this.<br />
<br />
* '''Feb 7''': 2.6 (the formulation of numerical evidence should be done by Sage if you've got Sage working, and by calculator if not; you can use an online tool like [http://primes.utm.edu/curios/includes/primetest.php this] to test whether a number is prime.) 2.8,2.9,2.11,2.12,2.14,2.19<br />
<br />
* '''Feb 17''': 2.15, 2.18, 2.23, 2.26, 2.30.<br />
<br />
Problem A: Prove that if n=pq, with p,q prime, then n is not a Carmichael number.<br />
<br />
* '''Feb 21''': Book problems: 3.4, 3.5, 3.6<br />
<br />
Problem A. Prove that there are infinitely many primes p such that 2 is '''not''' a primitive root in Z/pZ. We break this up into steps.<br />
<br />
Problem A.1. Prove that, if x is an element of Z/nZ, then x^2 is not a primitive root.<br />
<br />
Problem A.2. Prove that there are infinitely many primes p such that 2 is a square in Z/pZ. Hint: suppose there are only finitely many such primes p_1, .. p_r, and define N = (p_1 .. p_r)^2 - 2. Where can you go from here...?<br />
<br />
Problem A.3. Give a list of five primes p such that 2 is not a primitive root in Z/pZ (you can use the method of this proof or any other.)<br />
<br />
Problem B. Prove that 24 is the largest integer n such that every element of (Z/nZ)^* is a root of x^2-1.<br />
<br />
* '''Feb 28''': Problem A. Using p = 23 and q=31, show how to encrypt the message x=240 with Rabin's algorithm. Find all possible decryptions of your encrypted message.<br />
<br />
* '''March 23''': 4.1 from the book.<br />
<br />
Problem A. Give a prime factorization of the Gaussian integer 7+9i.<br />
<br />
Problem B. Read the notes from [http://www.math.ucsd.edu/~jverstra/Gaussian1.pdf here]. Note that Z[i] satisfies a ''reduction theorem'': if n and d are Gaussian integers, then there exists integers q and r such that n = qd + r and Norm(r) < Norm(d). But (by contrast with the case of Z) this d may not be unique. In some contexts it is better to be able to choose r uniquely, even if this means letting r have norm greater than Norm(d). <br />
<br />
B.1. When d = 1+2i, show that, for each n in Z[i], there is a '''unique''' pair (q,r) in Z[i] such that n = qd+r and r is contained in the set {0,1,2,3,4}. For instance, i can be written as i(1+2i) + 2, so we say i reduces to 2 mod (1+2i).<br />
(Hint: Suppose q(1+2i)+r = q'(1+2i) + r'. What can we say about (r-r'), and why is this incompatible with both r and r' being in {0,1,2,3,4}?<br />
<br />
B.2. Show that if n is an integer in Z, the reduction of n mod (1+2i) is equal to its reduction mod 5.<br />
<br />
Problem C. Let's try to figure out how to define "phi(d)" for a Gaussian integer d. Suppose S is a set of Gaussian integers such that every n in Z[i] can be written uniquely as qd+r, with q in Z[i] and r in S. (So for instance when d=1+2i, we showed in problem B that we can take S to be {0...4}. It would also be OK to take S to be {1..5} or {0,i,2i,1+i,1+2i}. In fact, it turns out that S has to be a set of size Norm(d) (I might or might not prove this in class; if not, feel free just to accept it.)<br />
<br />
Now define phi(d) to be the number of elements s of S such that s and d are coprime.<br />
<br />
C.1. Compute phi(1+2i) and phi(3).<br />
<br />
C.2. Prove that the value of phi(d) does not depend on the choice of S.<br />
<br />
C.3. Prove that every n in Z[i] which is coprime to 3 satisfies n^phi(3) = 1 mod 3; that is, Euler's theorem holds in this case. (You can prove this by direct computation; of course, if you want, you are welcome to prove that Euler's theorem holds for Z[i] in general, adapting the proof in Stein's book or the one we gave in class.)<br />
<br />
*'''March 30''' <br />
<br />
Problem A. Express 50005 as the sum of two squares.<br />
<br />
In the next two problems we denote by r(n) the number of ways to express n as the sum of two squares (i.e. the number of ordered pairs (a,b) such that a^2 + b^2 = n.) For instance, r(5) = 8: (a,b) = (+/-1, +/-2) or (+/-2, +/-1).<br />
<br />
Problem B. Prove that, for any N, there exists an integer n such that r(n) > N. (I.E., the function r(n) is ''unbounded.'')<br />
<br />
Problem C. If you like Sage, write a short program in Sage to compute r(n) and compute the average of r(n) as n ranges from 1 to 1000. Whether or not you like Sage, make a guess as to how this average would behave if you replaced "1000" by a larger and larger number. (Feel free to ask the Sage-lovers what answer they got in the optional first part of the question.) Can you prove this guess is correct?<br />
<br />
Problem D. We saw in class that the ring Z[sqrt(-5)] doesn't have unique factorization; 6 can be factored as 2*3 or (1+sqrt(-5))(1-sqrt(-5)). In this problem, we will prove that Z[sqrt(-d)] fails to have unique factorization for EVERY odd d >= 5. (Actually it's true for all d >= 5 but to make the proof manageable we'll restrict to the odd case.]<br />
<br />
D.1. Show that (1+sqrt(-d)), (1-sqrt(-d)) and 2 are irreducible in Z[sqrt(-d)].<br />
<br />
D.2. Now give an element of Z[sqrt(-d)] that has two distinct factorizations into irreducibles. (Hint: imitate the example we used for Z[sqrt(-5)].)<br />
''Remark:'' The rings Z[sqrt(d)], where d is positive, are quite different -- here we believe that there are infinitely many with unique factorization, though this conjecture has remained unproved for many decades!<br />
<br />
*'''April 17'''<br />
<br />
Book problems: 5.3,5.4,5.5<br />
<br />
Problem A: We discussed in class the problem of studying which positive integers are the sum of two squares. In this problem we prove that every element n of (Z/pZ) is the sum of two squares. We argue as follows. Let S be the set of squares in (Z/pZ), and let T be the set of (Z/pZ) consisting of all elements of the form n - x^2, for some x in (Z/pZ).<br />
<br />
A.1. What is the size of S and of T? Use this to show that S and T are not disjoint.<br />
<br />
A.2. Given that S and T are not disjoint, prove that n is the sum of two squares.<br />
<br />
Problem B. Using the continued fraction expansion, find a solution to the Pell equation x^2 - 13 y^2 = 1.<br />
<br />
Problem C. Show that the modified Pell equation x^2 - 7y^2 = -1 has no solutions in integers x,y. (Hint: reduce the equation modulo a suitably chosen prime.)<br />
<br />
Problem D. Stuffy Stirnweiss finished the 1945 season with a batting average of .3085443. Using continued fractions, guess how many at-bats he had. Tony Cuccinello had a batting average of .3084577. Given that he had more than 200 and fewer than 600 at-bats, can you estimate the number of at-bats he had?<br />
<br />
<!--<br />
'''December 7'''<br />
<br />
Problem A. When p is a prime congruent to 3 mod 4, prove that ((p-1)/2)! is either 1 or -1 in (Z/pZ). '''OPTIONAL:''' Use sage to compute whether this factorial is 1 or -1 for many primes p. Is there any pattern? Does it seem to be 1 half the time and -1 half the time? Note that I have no idea what the answer to this question is.<br />
<br />
Problem B. Using the continued fraction expansion, find a solution to the Pell equation x^2 - 13 y^2 = 1.<br />
<br />
Problem C. Show that the modified Pell equation x^2 - 7y^2 = -1 has no solutions in integers x,y. (Hint: reduce the equation modulo a suitably chosen prime.)<br />
<br />
Problem D.1. Pick two values of a in F_11 = Z/11Z (a not equal to 3), such that the equation y^2 = x^3+ax+1 defines an elliptic curve (i.e., it is smooth). For each such a, determine the number of points #E(F_11) and check that it falls inside the interval described in class. <br />
<br />
Problem D.2. The point P = (0,1) lies on each of these curves. For a =3 (the curve discussed in class) determine the order of P in the elliptic curve group, that is, find the smallest positive integer n such that nP = (infinity) -- the identity element in the group law.<br />
<br />
<!-- '''November 19'''<br />
<br />
Problem A. Using the method discussed in class (which is also the method of problem 4.6 in Stein, which in retrospect I think was too hard to assign with no preparation) find a nontrivial solution with y > 0 to the equation x^2 + 11y^2 = z^2. <br />
Problem B. Let f(X) be the number of solutions of A^2 + B^2 = C^3 such that A^2, B^2, and C^3 are all at most X. Use the heuristic described in class to explain why one might expect f(X) to grow more or less like X^{1/3}.<br />
<br />
Problem C. Show that if A+Bi is the cube of a Gaussian integer, then A^2 + B^2 is a perfect cube.<br />
<br />
Do ONE of problem D1 and D2; D1 is for people who like Sage, D2 is for people who like proving things.<br />
<br />
D1. Use Sage to compute f(X) for X = 1000, 10000, 100000. Does the answer look consistent with the heuristic prediction you made in B? Does f(X)/X^{1/3} appear to be approaching a limit?<br />
D2. We will use problem C to give a lower bound for f(X). Use this to show that there are at least X^{1/3} Gaussian integers with norm at most X that are perfect cubes. From here, show that f(X) > X^{1/3}.<br />
<br />
OPTIONAL: Show that the converse of problem C also holds -- A+Bi has norm a perfect cube if and only if A+Bi is a perfect cube. (You will need unique factorization of Gaussian integers.) Using this fact, prove that f(X) is bounded above by C X^{1/3} for some constant C.<br />
<br />
'''December 8 (note nonstandard due date)'''<br />
<br />
Problem A: Hyperbolas, ellipses, and "magic slopes"<br />
<br />
In the usual analytic geometry, both hyperbolas and ellipses are given by equations of the form<br />
<br />
ax^2 + bxy + cy^2 = d (*)<br />
<br />
<br />
(we always assume d is nonzero.)<br />
<br />
How can we tell whether such an equation describes an ellipse or a<br />
hyperbola? Well, in the geometric setting we all know and love (i.e. in<br />
R^2) a hyperbola has asymptotes and an ellipse does not. What is an <br />
asymptote? You could say it's "a line which doesn't intersect the curve", <br />
but ellipses have such lines too. You might want to say "a line which <br />
doesn't intersect the curve but comes closer and closer to the curve," but <br />
the problem here is that this a) is somewhat imprecise, and b) relies on a <br />
notion of "closer" that is not going to be very clear in Z/pZ, which is <br />
our ultimate goal!<br />
<br />
<br />
So let me state it a different way: let's say the asymptotes of a <br />
hyperbola have slopes m_1 and m_2. These are "magic slopes" in the <br />
following sense: ANY line of slope m_1 (and ditto for m_2) intersects the <br />
hyperbola in at most one point. An ellipse doesn't have "magic slopes" -- <br />
you can convince yourself by drawing pictures that, for any slope m, you <br />
can find a line of slope m striking the ellipse twice.<br />
<br />
<br />
[REMARK: For technical reasons we're going to ignore hyperbolae with a <br />
vertical asymptote, since vertical lines don't exactly have slope.]<br />
<br />
<br />
Now you might know the criterion that, over the real numbers.<br />
<br />
<br />
ax^2 + bxy + cy^2 = d<br />
<br />
<br />
is a hyperbola if b^2 - 4ac > 0 and an ellipse if b^2 - 4ac < 0. (For <br />
instance, x^2 - y^2 = 1 is a hyperbola with magic slopes 1 and -1, and x^2 <br />
+ y^2 = 1 is an ellipse.) <br />
This criterion does not make sense in Z/pZ, where there is no notion of <br />
"greater" or "less."<br />
<br />
<br />
QUESTION A.1.<br />
<br />
For which primes p is it the case that <br />
<br />
x^2 + y^2 = 1<br />
<br />
has magic slopes in Z/pZ? <br />
<br />
QUESTION A.2. For which primes p is it the case that<br />
<br />
<br />
x^2 - y^2 = 1<br />
<br />
<br />
has magic slopes in Z/pZ?<br />
<br />
<br />
QUESTION B: Back in the very first week of this course we studied the set of integers x such that x^2 + 1 is prime. Based on the heuristics we discussed in class, about how many x between 1 and N would you expect to satisfy "x^2 + 1 is prime?"<br />
<br />
<br />
QUESTION C: Let z be a complex number. Then the set of all complex numbers of the form a + bz, with a and b integers, forms a lattice in the complex plane (thought of as R^2.) What is the covolume of this lattice, in terms of z?<br />
<br />
<br />
QUESTION D: Let z be a complex number. Using Minkowski's theorem and the result of question C, show that there exist integers a,b, not both zero, such that Norm(a+bz) is at most (4/pi) Im(z). (Hint: let Omega be the set of all complex numbers of norm at most (4/pi) Im(z).)<br />
<br />
<br />
OPTIONAL EXTRA: Can you give an exact formula for the minimal norm of any nonzero complex number of the form a+bz?<br />
<br />
'''December 12''' Book problems 6.1, 6.2, 6.5, 6.10. For extra credit attempt to do problem 6.9.<br />
<br />
--></div>Andreichttps://www.math.wisc.edu/wiki/index.php?title=Math_567_--_Elementary_Number_Theory&diff=19351Math 567 -- Elementary Number Theory2020-04-13T04:50:37Z<p>Andreic: </p>
<hr />
<div>'''MATH 567<br />
<br />
Elementary Number Theory'''<br />
<br />
MWF 1:20-2:10, Van Vleck B119<br />
<br />
'''Professor:''' [http://www.math.wisc.edu/~andreic/ Andrei Caldararu] (andreic@math.wisc.edu)<br />
<br />
''Office Hours:'' Wednesdays 2:30-3:30, Van Vleck 605. <br />
<br />
'''Grader:''' Yifan Peng, email: peng64@wisc.edu<br />
<br />
''Office Hours:'' TBA.<br />
<br />
Math 567 is a course in elementary number theory, aimed at undergraduates majoring in math or other quantitative disciplines. A general familiarity with abstract algebra at the level of Math 541 will be assumed, but students who haven't taken 541 are welcome to attend if they're willing to play a little catchup. We will be using William Stein's new (and cheap) textbook [http://www.amazon.com/Elementary-Number-Theory-Computational-Undergraduate/dp/0387855246 Elementary Number Theory: Primes, Congruences, and Secrets], which emphasizes computational approaches to the subject. If you don't need a physical copy of the book, [http://www.williamstein.org/ent/ it is available as a free legal .pdf.] We will be using the (free, public-domain) mathematical software [http://www.sagemath.org/ SAGE], developed largely by Stein, as an integral component of our coursework. There is a [http://doc.sagemath.org/pdf/en/tutorial/SageTutorial.pdf useful online tutorial.] You can download SAGE to your own computer or [http://www.sagenb.org use it online].<br />
<br />
Topics include some subset of, but are not limited to: Divisibility, the Euclidean algorithm and the GCD, linear Diophantine equations, prime numbers and uniqueness of factorization. Congruences, Chinese remainder theorem, Fermat's "little" theorem, Wilson's theorem, Euler's theorem and totient function, the RSA cryptosystem, Rabin's encryption scheme, Diffie-Hellman key exchange protocol. Number-theoretic functions, multiplicative functions, Möbius inversion. Primitive roots and indices. Quadratic reciprocity and the Legendre symbol. Perfect numbers, Mersenne primes, Fermat primes. Pythagorean triples, Special cases of Fermat's "last" theorem. Fibonacci numbers. Continued fractions. Distribution of primes, discussion of prime number theorem. Primality testing and factoring algorithms.<br />
<br />
'''Course Policies:''' Homework will be due on Fridays. It can be turned in late only with ''advance'' permission from your grader. It is acceptable to use calculators and computers on homework (indeed, some of it will require a computer) but calculators are not allowed during exams. You are encouraged to work together on homework, but writeups must be done individually.<br />
<br />
Many of the problems in this course will ask you to prove things. I expect proofs to be written in English sentences; the proofs in Stein's book are a good model for the level of verbosity I am looking for. <br />
<br />
'''Grading:''' The grade in Math 567 will be composed of 50% homework, 25% midterm, 25% final. The midterm will be on March 11, in class. The final exam will be on 5/8/2020, 7:45AM-9:45AM in room TBA.<br />
<br />
'''Syllabus:''' <br />
(This may change as we see what pace works well for the course. All section numbers refer to Stein's book.)<br />
<br />
* Jan 22-31: Prime numbers, prime factorizations, Euclidean algorithm and GCD (1.1-1.2)<br />
* Feb 3-7: The integers mod n, Euler's theorem, the phi function (2.1-2.2)<br />
* Feb 10-14: Modular exponentiation, primality testing, and primitive roots (2.4-2.5)<br />
* Feb 17-21: Public-key cryptography and RSA (3.1-3.4)<br />
* Feb 24-28: Rabin's algorithm (not in the book); algebraic numbers <br />
* Mar 2-6: Quadratic reciprocity (4.1-4.4)<br />
* Mar 9, Mar 13: Finite and infinite continued fractions (5.1-5.3)<br />
* Mar 11: ''Midterm exam''<br />
* Mar 23-30: Continued fractions and diophantine approximation (5.4-5.5)<br />
* Apr 1-3: Diophantine equations I: Pell's equation and Lagrange's theorem<br />
* Apr 6-10: More diophantine equations, elliptic curves (6.1)<br />
* Apr 13-17: Applications of elliptic curves (6.2-6.3)<br />
* Apr 20-24: More applications of elliptic curves (6.4)<br />
* Apr 27-May 1: advanced topic TBD: maybe additional discussion of cryptographic techniques?<br />
<br />
'''Video Lectures:'''<br />
Due to the university being transitioned to online teaching only, out lectures after spring break will need to be delivered online. I will post here videos for you to watch for the material we need to cover. Please watch them, and please give me feedback on how they could be improved.<br />
<br />
* '''Lecture for Mar. 23''': watch it [https://mediaspace.wisc.edu/media/Math+567+March+23+Lecture/0_fkhvkxxv here].<br />
* '''Lecture for Mar. 25''': watch it [https://mediaspace.wisc.edu/media/Math+567+March+25+lecture/0_qgkhpm19 here].<br />
* '''Lecture for Mar. 27''': watch it [https://mediaspace.wisc.edu/media/March+27+Math+567+Lecture+Video/0_ajbr2ja0 here].<br />
* '''Lecture for Mar. 30''': watch it [https://mediaspace.wisc.edu/media/March+30+Math+567+Lecture/0_yz44sw2c here].<br />
* '''Lecture for Apr. 1''': watch it [https://mediaspace.wisc.edu/media/April+1+Math+567/0_e64bdfcf here].<br />
* '''Lecture for Apr. 3''': watch it [https://mediaspace.wisc.edu/media/April+3+Math+567+Video/0_02rvfino here].<br />
* '''Lecture for Apr. 6''': watch it [https://mediaspace.wisc.edu/media/April+6+Math+567+Video/1_jmm8r268 here].<br />
* '''Lecture for Apr. 8''': watch it [https://mediaspace.wisc.edu/media/Group+law+elliptic/1_os81ndwu here].<br />
* '''Lecture for Apr. 10''': watch it [https://mediaspace.wisc.edu/media/Pollard%27s+p-1+method+for+factoring/1_qyf08tuv here].<br />
* '''Lecture for Apr. 13''': watch it [https://mediaspace.wisc.edu/media/Lenstra%2C+El+Gamal/1_848flqos here].<br />
<br />
'''Homework:'''<br />
Homework is due at the beginning of class on the specified Friday. Typing your homework is not a requirement, but if you don't already know LaTeX I highly recommend that you learn it and use it to typeset your homework. I will sometimes assign extra problems, which I will e-mail to the class list and include here.<br />
<br />
* '''Jan 31''': 1.1, 1.3, 1.5, 1.7 (use SAGE), 1.8, 1.14.<br />
Problem A: Use SAGE to compute the number of x in [1..N] such that x^2 + 1 is prime, for N = 100, N = 1000, and N = 10000. Let f(N) be the number of such x. <br />
<br />
a) Can you formulate a conjecture about the relationship between f(N) and N/log N? <br />
<br />
b) What if x^2 + 1 is replaced with x^2 + 2? Can you explain why x^2 + 2 appears less likely to be prime? (Hint: consider x mod 3.)<br />
<br />
c) Prove that f(N) is at most (1/2)N+1. (Hint: consider x mod 2.) <br />
<br />
d) Give as good an upper bound as you can for f(N).<br />
<br />
Note that, despite the evident regularities you'll observe in this problem, we do not even know whether there are infinitely many primes of the form x^2 + 1! You would become very famous if you proved this.<br />
<br />
* '''Feb 7''': 2.6 (the formulation of numerical evidence should be done by Sage if you've got Sage working, and by calculator if not; you can use an online tool like [http://primes.utm.edu/curios/includes/primetest.php this] to test whether a number is prime.) 2.8,2.9,2.11,2.12,2.14,2.19<br />
<br />
* '''Feb 17''': 2.15, 2.18, 2.23, 2.26, 2.30.<br />
<br />
Problem A: Prove that if n=pq, with p,q prime, then n is not a Carmichael number.<br />
<br />
* '''Feb 21''': Book problems: 3.4, 3.5, 3.6<br />
<br />
Problem A. Prove that there are infinitely many primes p such that 2 is '''not''' a primitive root in Z/pZ. We break this up into steps.<br />
<br />
Problem A.1. Prove that, if x is an element of Z/nZ, then x^2 is not a primitive root.<br />
<br />
Problem A.2. Prove that there are infinitely many primes p such that 2 is a square in Z/pZ. Hint: suppose there are only finitely many such primes p_1, .. p_r, and define N = (p_1 .. p_r)^2 - 2. Where can you go from here...?<br />
<br />
Problem A.3. Give a list of five primes p such that 2 is not a primitive root in Z/pZ (you can use the method of this proof or any other.)<br />
<br />
Problem B. Prove that 24 is the largest integer n such that every element of (Z/nZ)^* is a root of x^2-1.<br />
<br />
* '''Feb 28''': Problem A. Using p = 23 and q=31, show how to encrypt the message x=240 with Rabin's algorithm. Find all possible decryptions of your encrypted message.<br />
<br />
* '''March 23''': 4.1 from the book.<br />
<br />
Problem A. Give a prime factorization of the Gaussian integer 7+9i.<br />
<br />
Problem B. Read the notes from [http://www.math.ucsd.edu/~jverstra/Gaussian1.pdf here]. Note that Z[i] satisfies a ''reduction theorem'': if n and d are Gaussian integers, then there exists integers q and r such that n = qd + r and Norm(r) < Norm(d). But (by contrast with the case of Z) this d may not be unique. In some contexts it is better to be able to choose r uniquely, even if this means letting r have norm greater than Norm(d). <br />
<br />
B.1. When d = 1+2i, show that, for each n in Z[i], there is a '''unique''' pair (q,r) in Z[i] such that n = qd+r and r is contained in the set {0,1,2,3,4}. For instance, i can be written as i(1+2i) + 2, so we say i reduces to 2 mod (1+2i).<br />
(Hint: Suppose q(1+2i)+r = q'(1+2i) + r'. What can we say about (r-r'), and why is this incompatible with both r and r' being in {0,1,2,3,4}?<br />
<br />
B.2. Show that if n is an integer in Z, the reduction of n mod (1+2i) is equal to its reduction mod 5.<br />
<br />
Problem C. Let's try to figure out how to define "phi(d)" for a Gaussian integer d. Suppose S is a set of Gaussian integers such that every n in Z[i] can be written uniquely as qd+r, with q in Z[i] and r in S. (So for instance when d=1+2i, we showed in problem B that we can take S to be {0...4}. It would also be OK to take S to be {1..5} or {0,i,2i,1+i,1+2i}. In fact, it turns out that S has to be a set of size Norm(d) (I might or might not prove this in class; if not, feel free just to accept it.)<br />
<br />
Now define phi(d) to be the number of elements s of S such that s and d are coprime.<br />
<br />
C.1. Compute phi(1+2i) and phi(3).<br />
<br />
C.2. Prove that the value of phi(d) does not depend on the choice of S.<br />
<br />
C.3. Prove that every n in Z[i] which is coprime to 3 satisfies n^phi(3) = 1 mod 3; that is, Euler's theorem holds in this case. (You can prove this by direct computation; of course, if you want, you are welcome to prove that Euler's theorem holds for Z[i] in general, adapting the proof in Stein's book or the one we gave in class.)<br />
<br />
*'''March 30''' <br />
<br />
Problem A. Express 50005 as the sum of two squares.<br />
<br />
In the next two problems we denote by r(n) the number of ways to express n as the sum of two squares (i.e. the number of ordered pairs (a,b) such that a^2 + b^2 = n.) For instance, r(5) = 8: (a,b) = (+/-1, +/-2) or (+/-2, +/-1).<br />
<br />
Problem B. Prove that, for any N, there exists an integer n such that r(n) > N. (I.E., the function r(n) is ''unbounded.'')<br />
<br />
Problem C. If you like Sage, write a short program in Sage to compute r(n) and compute the average of r(n) as n ranges from 1 to 1000. Whether or not you like Sage, make a guess as to how this average would behave if you replaced "1000" by a larger and larger number. (Feel free to ask the Sage-lovers what answer they got in the optional first part of the question.) Can you prove this guess is correct?<br />
<br />
Problem D. We saw in class that the ring Z[sqrt(-5)] doesn't have unique factorization; 6 can be factored as 2*3 or (1+sqrt(-5))(1-sqrt(-5)). In this problem, we will prove that Z[sqrt(-d)] fails to have unique factorization for EVERY odd d >= 5. (Actually it's true for all d >= 5 but to make the proof manageable we'll restrict to the odd case.]<br />
<br />
D.1. Show that (1+sqrt(-d)), (1-sqrt(-d)) and 2 are irreducible in Z[sqrt(-d)].<br />
<br />
D.2. Now give an element of Z[sqrt(-d)] that has two distinct factorizations into irreducibles. (Hint: imitate the example we used for Z[sqrt(-5)].)<br />
''Remark:'' The rings Z[sqrt(d)], where d is positive, are quite different -- here we believe that there are infinitely many with unique factorization, though this conjecture has remained unproved for many decades!<br />
<br />
*'''April 17'''<br />
<br />
Book problems: 5.3,5.4,5.5<br />
<br />
Problem A: We discussed in class the problem of studying which positive integers are the sum of two squares. In this problem we prove that every element n of (Z/pZ) is the sum of two squares. We argue as follows. Let S be the set of squares in (Z/pZ), and let T be the set of (Z/pZ) consisting of all elements of the form n - x^2, for some x in (Z/pZ).<br />
<br />
A.1. What is the size of S and of T? Use this to show that S and T are not disjoint.<br />
<br />
A.2. Given that S and T are not disjoint, prove that n is the sum of two squares.<br />
<br />
Problem B. Using the continued fraction expansion, find a solution to the Pell equation x^2 - 13 y^2 = 1.<br />
<br />
Problem C. Show that the modified Pell equation x^2 - 7y^2 = -1 has no solutions in integers x,y. (Hint: reduce the equation modulo a suitably chosen prime.)<br />
<br />
Problem D. Stuffy Stirnweiss finished the 1945 season with a batting average of .3085443. Using continued fractions, guess how many at-bats he had. Tony Cuccinello had a batting average of .3084577. Given that he had more than 200 and fewer than 600 at-bats, can you estimate the number of at-bats he had?<br />
<br />
<!--<br />
'''December 7'''<br />
<br />
Problem A. When p is a prime congruent to 3 mod 4, prove that ((p-1)/2)! is either 1 or -1 in (Z/pZ). '''OPTIONAL:''' Use sage to compute whether this factorial is 1 or -1 for many primes p. Is there any pattern? Does it seem to be 1 half the time and -1 half the time? Note that I have no idea what the answer to this question is.<br />
<br />
Problem B. Using the continued fraction expansion, find a solution to the Pell equation x^2 - 13 y^2 = 1.<br />
<br />
Problem C. Show that the modified Pell equation x^2 - 7y^2 = -1 has no solutions in integers x,y. (Hint: reduce the equation modulo a suitably chosen prime.)<br />
<br />
Problem D.1. Pick two values of a in F_11 = Z/11Z (a not equal to 3), such that the equation y^2 = x^3+ax+1 defines an elliptic curve (i.e., it is smooth). For each such a, determine the number of points #E(F_11) and check that it falls inside the interval described in class. <br />
<br />
Problem D.2. The point P = (0,1) lies on each of these curves. For a =3 (the curve discussed in class) determine the order of P in the elliptic curve group, that is, find the smallest positive integer n such that nP = (infinity) -- the identity element in the group law.<br />
<br />
<!-- '''November 19'''<br />
<br />
Problem A. Using the method discussed in class (which is also the method of problem 4.6 in Stein, which in retrospect I think was too hard to assign with no preparation) find a nontrivial solution with y > 0 to the equation x^2 + 11y^2 = z^2. <br />
Problem B. Let f(X) be the number of solutions of A^2 + B^2 = C^3 such that A^2, B^2, and C^3 are all at most X. Use the heuristic described in class to explain why one might expect f(X) to grow more or less like X^{1/3}.<br />
<br />
Problem C. Show that if A+Bi is the cube of a Gaussian integer, then A^2 + B^2 is a perfect cube.<br />
<br />
Do ONE of problem D1 and D2; D1 is for people who like Sage, D2 is for people who like proving things.<br />
<br />
D1. Use Sage to compute f(X) for X = 1000, 10000, 100000. Does the answer look consistent with the heuristic prediction you made in B? Does f(X)/X^{1/3} appear to be approaching a limit?<br />
D2. We will use problem C to give a lower bound for f(X). Use this to show that there are at least X^{1/3} Gaussian integers with norm at most X that are perfect cubes. From here, show that f(X) > X^{1/3}.<br />
<br />
OPTIONAL: Show that the converse of problem C also holds -- A+Bi has norm a perfect cube if and only if A+Bi is a perfect cube. (You will need unique factorization of Gaussian integers.) Using this fact, prove that f(X) is bounded above by C X^{1/3} for some constant C.<br />
<br />
'''December 8 (note nonstandard due date)'''<br />
<br />
Problem A: Hyperbolas, ellipses, and "magic slopes"<br />
<br />
In the usual analytic geometry, both hyperbolas and ellipses are given by equations of the form<br />
<br />
ax^2 + bxy + cy^2 = d (*)<br />
<br />
<br />
(we always assume d is nonzero.)<br />
<br />
How can we tell whether such an equation describes an ellipse or a<br />
hyperbola? Well, in the geometric setting we all know and love (i.e. in<br />
R^2) a hyperbola has asymptotes and an ellipse does not. What is an <br />
asymptote? You could say it's "a line which doesn't intersect the curve", <br />
but ellipses have such lines too. You might want to say "a line which <br />
doesn't intersect the curve but comes closer and closer to the curve," but <br />
the problem here is that this a) is somewhat imprecise, and b) relies on a <br />
notion of "closer" that is not going to be very clear in Z/pZ, which is <br />
our ultimate goal!<br />
<br />
<br />
So let me state it a different way: let's say the asymptotes of a <br />
hyperbola have slopes m_1 and m_2. These are "magic slopes" in the <br />
following sense: ANY line of slope m_1 (and ditto for m_2) intersects the <br />
hyperbola in at most one point. An ellipse doesn't have "magic slopes" -- <br />
you can convince yourself by drawing pictures that, for any slope m, you <br />
can find a line of slope m striking the ellipse twice.<br />
<br />
<br />
[REMARK: For technical reasons we're going to ignore hyperbolae with a <br />
vertical asymptote, since vertical lines don't exactly have slope.]<br />
<br />
<br />
Now you might know the criterion that, over the real numbers.<br />
<br />
<br />
ax^2 + bxy + cy^2 = d<br />
<br />
<br />
is a hyperbola if b^2 - 4ac > 0 and an ellipse if b^2 - 4ac < 0. (For <br />
instance, x^2 - y^2 = 1 is a hyperbola with magic slopes 1 and -1, and x^2 <br />
+ y^2 = 1 is an ellipse.) <br />
This criterion does not make sense in Z/pZ, where there is no notion of <br />
"greater" or "less."<br />
<br />
<br />
QUESTION A.1.<br />
<br />
For which primes p is it the case that <br />
<br />
x^2 + y^2 = 1<br />
<br />
has magic slopes in Z/pZ? <br />
<br />
QUESTION A.2. For which primes p is it the case that<br />
<br />
<br />
x^2 - y^2 = 1<br />
<br />
<br />
has magic slopes in Z/pZ?<br />
<br />
<br />
QUESTION B: Back in the very first week of this course we studied the set of integers x such that x^2 + 1 is prime. Based on the heuristics we discussed in class, about how many x between 1 and N would you expect to satisfy "x^2 + 1 is prime?"<br />
<br />
<br />
QUESTION C: Let z be a complex number. Then the set of all complex numbers of the form a + bz, with a and b integers, forms a lattice in the complex plane (thought of as R^2.) What is the covolume of this lattice, in terms of z?<br />
<br />
<br />
QUESTION D: Let z be a complex number. Using Minkowski's theorem and the result of question C, show that there exist integers a,b, not both zero, such that Norm(a+bz) is at most (4/pi) Im(z). (Hint: let Omega be the set of all complex numbers of norm at most (4/pi) Im(z).)<br />
<br />
<br />
OPTIONAL EXTRA: Can you give an exact formula for the minimal norm of any nonzero complex number of the form a+bz?<br />
<br />
'''December 12''' Book problems 6.1, 6.2, 6.5, 6.10. For extra credit attempt to do problem 6.9.<br />
<br />
--></div>Andreichttps://www.math.wisc.edu/wiki/index.php?title=Math_567_--_Elementary_Number_Theory&diff=19348Math 567 -- Elementary Number Theory2020-04-10T20:16:00Z<p>Andreic: </p>
<hr />
<div>'''MATH 567<br />
<br />
Elementary Number Theory'''<br />
<br />
MWF 1:20-2:10, Van Vleck B119<br />
<br />
'''Professor:''' [http://www.math.wisc.edu/~andreic/ Andrei Caldararu] (andreic@math.wisc.edu)<br />
<br />
''Office Hours:'' Wednesdays 2:30-3:30, Van Vleck 605. <br />
<br />
'''Grader:''' Yifan Peng, email: peng64@wisc.edu<br />
<br />
''Office Hours:'' TBA.<br />
<br />
Math 567 is a course in elementary number theory, aimed at undergraduates majoring in math or other quantitative disciplines. A general familiarity with abstract algebra at the level of Math 541 will be assumed, but students who haven't taken 541 are welcome to attend if they're willing to play a little catchup. We will be using William Stein's new (and cheap) textbook [http://www.amazon.com/Elementary-Number-Theory-Computational-Undergraduate/dp/0387855246 Elementary Number Theory: Primes, Congruences, and Secrets], which emphasizes computational approaches to the subject. If you don't need a physical copy of the book, [http://www.williamstein.org/ent/ it is available as a free legal .pdf.] We will be using the (free, public-domain) mathematical software [http://www.sagemath.org/ SAGE], developed largely by Stein, as an integral component of our coursework. There is a [http://doc.sagemath.org/pdf/en/tutorial/SageTutorial.pdf useful online tutorial.] You can download SAGE to your own computer or [http://www.sagenb.org use it online].<br />
<br />
Topics include some subset of, but are not limited to: Divisibility, the Euclidean algorithm and the GCD, linear Diophantine equations, prime numbers and uniqueness of factorization. Congruences, Chinese remainder theorem, Fermat's "little" theorem, Wilson's theorem, Euler's theorem and totient function, the RSA cryptosystem, Rabin's encryption scheme, Diffie-Hellman key exchange protocol. Number-theoretic functions, multiplicative functions, Möbius inversion. Primitive roots and indices. Quadratic reciprocity and the Legendre symbol. Perfect numbers, Mersenne primes, Fermat primes. Pythagorean triples, Special cases of Fermat's "last" theorem. Fibonacci numbers. Continued fractions. Distribution of primes, discussion of prime number theorem. Primality testing and factoring algorithms.<br />
<br />
'''Course Policies:''' Homework will be due on Fridays. It can be turned in late only with ''advance'' permission from your grader. It is acceptable to use calculators and computers on homework (indeed, some of it will require a computer) but calculators are not allowed during exams. You are encouraged to work together on homework, but writeups must be done individually.<br />
<br />
Many of the problems in this course will ask you to prove things. I expect proofs to be written in English sentences; the proofs in Stein's book are a good model for the level of verbosity I am looking for. <br />
<br />
'''Grading:''' The grade in Math 567 will be composed of 50% homework, 25% midterm, 25% final. The midterm will be on March 11, in class. The final exam will be on 5/8/2020, 7:45AM-9:45AM in room TBA.<br />
<br />
'''Syllabus:''' <br />
(This may change as we see what pace works well for the course. All section numbers refer to Stein's book.)<br />
<br />
* Jan 22-31: Prime numbers, prime factorizations, Euclidean algorithm and GCD (1.1-1.2)<br />
* Feb 3-7: The integers mod n, Euler's theorem, the phi function (2.1-2.2)<br />
* Feb 10-14: Modular exponentiation, primality testing, and primitive roots (2.4-2.5)<br />
* Feb 17-21: Public-key cryptography and RSA (3.1-3.4)<br />
* Feb 24-28: Rabin's algorithm (not in the book); algebraic numbers <br />
* Mar 2-6: Quadratic reciprocity (4.1-4.4)<br />
* Mar 9, Mar 13: Finite and infinite continued fractions (5.1-5.3)<br />
* Mar 11: ''Midterm exam''<br />
* Mar 23-30: Continued fractions and diophantine approximation (5.4-5.5)<br />
* Apr 1-3: Diophantine equations I: Pell's equation and Lagrange's theorem<br />
* Apr 6-10: More diophantine equations, elliptic curves (6.1)<br />
* Apr 13-17: Applications of elliptic curves (6.2-6.3)<br />
* Apr 20-24: More applications of elliptic curves (6.4)<br />
* Apr 27-May 1: advanced topic TBD: maybe additional discussion of cryptographic techniques?<br />
<br />
'''Video Lectures:'''<br />
Due to the university being transitioned to online teaching only, out lectures after spring break will need to be delivered online. I will post here videos for you to watch for the material we need to cover. Please watch them, and please give me feedback on how they could be improved.<br />
<br />
* '''Lecture for Mar. 23''': watch it [https://mediaspace.wisc.edu/media/Math+567+March+23+Lecture/0_fkhvkxxv here].<br />
* '''Lecture for Mar. 25''': watch it [https://mediaspace.wisc.edu/media/Math+567+March+25+lecture/0_qgkhpm19 here].<br />
* '''Lecture for Mar. 27''': watch it [https://mediaspace.wisc.edu/media/March+27+Math+567+Lecture+Video/0_ajbr2ja0 here].<br />
* '''Lecture for Mar. 30''': watch it [https://mediaspace.wisc.edu/media/March+30+Math+567+Lecture/0_yz44sw2c here].<br />
* '''Lecture for Apr. 1''': watch it [https://mediaspace.wisc.edu/media/April+1+Math+567/0_e64bdfcf here].<br />
* '''Lecture for Apr. 3''': watch it [https://mediaspace.wisc.edu/media/April+3+Math+567+Video/0_02rvfino here].<br />
* '''Lecture for Apr. 6''': watch it [https://mediaspace.wisc.edu/media/April+6+Math+567+Video/1_jmm8r268 here].<br />
* '''Lecture for Apr. 8''': watch it [https://mediaspace.wisc.edu/media/Group+law+elliptic/1_os81ndwu here].<br />
* '''Lecture for Apr. 10''': watch it [https://mediaspace.wisc.edu/media/Pollard%27s+p-1+method+for+factoring/1_qyf08tuv here].<br />
<br />
'''Homework:'''<br />
Homework is due at the beginning of class on the specified Friday. Typing your homework is not a requirement, but if you don't already know LaTeX I highly recommend that you learn it and use it to typeset your homework. I will sometimes assign extra problems, which I will e-mail to the class list and include here.<br />
<br />
* '''Jan 31''': 1.1, 1.3, 1.5, 1.7 (use SAGE), 1.8, 1.14.<br />
Problem A: Use SAGE to compute the number of x in [1..N] such that x^2 + 1 is prime, for N = 100, N = 1000, and N = 10000. Let f(N) be the number of such x. <br />
<br />
a) Can you formulate a conjecture about the relationship between f(N) and N/log N? <br />
<br />
b) What if x^2 + 1 is replaced with x^2 + 2? Can you explain why x^2 + 2 appears less likely to be prime? (Hint: consider x mod 3.)<br />
<br />
c) Prove that f(N) is at most (1/2)N+1. (Hint: consider x mod 2.) <br />
<br />
d) Give as good an upper bound as you can for f(N).<br />
<br />
Note that, despite the evident regularities you'll observe in this problem, we do not even know whether there are infinitely many primes of the form x^2 + 1! You would become very famous if you proved this.<br />
<br />
* '''Feb 7''': 2.6 (the formulation of numerical evidence should be done by Sage if you've got Sage working, and by calculator if not; you can use an online tool like [http://primes.utm.edu/curios/includes/primetest.php this] to test whether a number is prime.) 2.8,2.9,2.11,2.12,2.14,2.19<br />
<br />
* '''Feb 17''': 2.15, 2.18, 2.23, 2.26, 2.30.<br />
<br />
Problem A: Prove that if n=pq, with p,q prime, then n is not a Carmichael number.<br />
<br />
* '''Feb 21''': Book problems: 3.4, 3.5, 3.6<br />
<br />
Problem A. Prove that there are infinitely many primes p such that 2 is '''not''' a primitive root in Z/pZ. We break this up into steps.<br />
<br />
Problem A.1. Prove that, if x is an element of Z/nZ, then x^2 is not a primitive root.<br />
<br />
Problem A.2. Prove that there are infinitely many primes p such that 2 is a square in Z/pZ. Hint: suppose there are only finitely many such primes p_1, .. p_r, and define N = (p_1 .. p_r)^2 - 2. Where can you go from here...?<br />
<br />
Problem A.3. Give a list of five primes p such that 2 is not a primitive root in Z/pZ (you can use the method of this proof or any other.)<br />
<br />
Problem B. Prove that 24 is the largest integer n such that every element of (Z/nZ)^* is a root of x^2-1.<br />
<br />
* '''Feb 28''': Problem A. Using p = 23 and q=31, show how to encrypt the message x=240 with Rabin's algorithm. Find all possible decryptions of your encrypted message.<br />
<br />
* '''March 23''': 4.1 from the book.<br />
<br />
Problem A. Give a prime factorization of the Gaussian integer 7+9i.<br />
<br />
Problem B. Read the notes from [http://www.math.ucsd.edu/~jverstra/Gaussian1.pdf here]. Note that Z[i] satisfies a ''reduction theorem'': if n and d are Gaussian integers, then there exists integers q and r such that n = qd + r and Norm(r) < Norm(d). But (by contrast with the case of Z) this d may not be unique. In some contexts it is better to be able to choose r uniquely, even if this means letting r have norm greater than Norm(d). <br />
<br />
B.1. When d = 1+2i, show that, for each n in Z[i], there is a '''unique''' pair (q,r) in Z[i] such that n = qd+r and r is contained in the set {0,1,2,3,4}. For instance, i can be written as i(1+2i) + 2, so we say i reduces to 2 mod (1+2i).<br />
(Hint: Suppose q(1+2i)+r = q'(1+2i) + r'. What can we say about (r-r'), and why is this incompatible with both r and r' being in {0,1,2,3,4}?<br />
<br />
B.2. Show that if n is an integer in Z, the reduction of n mod (1+2i) is equal to its reduction mod 5.<br />
<br />
Problem C. Let's try to figure out how to define "phi(d)" for a Gaussian integer d. Suppose S is a set of Gaussian integers such that every n in Z[i] can be written uniquely as qd+r, with q in Z[i] and r in S. (So for instance when d=1+2i, we showed in problem B that we can take S to be {0...4}. It would also be OK to take S to be {1..5} or {0,i,2i,1+i,1+2i}. In fact, it turns out that S has to be a set of size Norm(d) (I might or might not prove this in class; if not, feel free just to accept it.)<br />
<br />
Now define phi(d) to be the number of elements s of S such that s and d are coprime.<br />
<br />
C.1. Compute phi(1+2i) and phi(3).<br />
<br />
C.2. Prove that the value of phi(d) does not depend on the choice of S.<br />
<br />
C.3. Prove that every n in Z[i] which is coprime to 3 satisfies n^phi(3) = 1 mod 3; that is, Euler's theorem holds in this case. (You can prove this by direct computation; of course, if you want, you are welcome to prove that Euler's theorem holds for Z[i] in general, adapting the proof in Stein's book or the one we gave in class.)<br />
<br />
*'''March 30''' <br />
<br />
Problem A. Express 50005 as the sum of two squares.<br />
<br />
In the next two problems we denote by r(n) the number of ways to express n as the sum of two squares (i.e. the number of ordered pairs (a,b) such that a^2 + b^2 = n.) For instance, r(5) = 8: (a,b) = (+/-1, +/-2) or (+/-2, +/-1).<br />
<br />
Problem B. Prove that, for any N, there exists an integer n such that r(n) > N. (I.E., the function r(n) is ''unbounded.'')<br />
<br />
Problem C. If you like Sage, write a short program in Sage to compute r(n) and compute the average of r(n) as n ranges from 1 to 1000. Whether or not you like Sage, make a guess as to how this average would behave if you replaced "1000" by a larger and larger number. (Feel free to ask the Sage-lovers what answer they got in the optional first part of the question.) Can you prove this guess is correct?<br />
<br />
Problem D. We saw in class that the ring Z[sqrt(-5)] doesn't have unique factorization; 6 can be factored as 2*3 or (1+sqrt(-5))(1-sqrt(-5)). In this problem, we will prove that Z[sqrt(-d)] fails to have unique factorization for EVERY odd d >= 5. (Actually it's true for all d >= 5 but to make the proof manageable we'll restrict to the odd case.]<br />
<br />
D.1. Show that (1+sqrt(-d)), (1-sqrt(-d)) and 2 are irreducible in Z[sqrt(-d)].<br />
<br />
D.2. Now give an element of Z[sqrt(-d)] that has two distinct factorizations into irreducibles. (Hint: imitate the example we used for Z[sqrt(-5)].)<br />
''Remark:'' The rings Z[sqrt(d)], where d is positive, are quite different -- here we believe that there are infinitely many with unique factorization, though this conjecture has remained unproved for many decades!<br />
<br />
*'''April 17'''<br />
<br />
Book problems: 5.3,5.4,5.5<br />
<br />
Problem A: We discussed in class the problem of studying which positive integers are the sum of two squares. In this problem we prove that every element n of (Z/pZ) is the sum of two squares. We argue as follows. Let S be the set of squares in (Z/pZ), and let T be the set of (Z/pZ) consisting of all elements of the form n - x^2, for some x in (Z/pZ).<br />
<br />
A.1. What is the size of S and of T? Use this to show that S and T are not disjoint.<br />
<br />
A.2. Given that S and T are not disjoint, prove that n is the sum of two squares.<br />
<br />
Problem B. Using the continued fraction expansion, find a solution to the Pell equation x^2 - 13 y^2 = 1.<br />
<br />
Problem C. Show that the modified Pell equation x^2 - 7y^2 = -1 has no solutions in integers x,y. (Hint: reduce the equation modulo a suitably chosen prime.)<br />
<br />
Problem D. Stuffy Stirnweiss finished the 1945 season with a batting average of .3085443. Using continued fractions, guess how many at-bats he had. Tony Cuccinello had a batting average of .3084577. Given that he had more than 200 and fewer than 600 at-bats, can you estimate the number of at-bats he had?<br />
<br />
<!--<br />
'''December 7'''<br />
<br />
Problem A. When p is a prime congruent to 3 mod 4, prove that ((p-1)/2)! is either 1 or -1 in (Z/pZ). '''OPTIONAL:''' Use sage to compute whether this factorial is 1 or -1 for many primes p. Is there any pattern? Does it seem to be 1 half the time and -1 half the time? Note that I have no idea what the answer to this question is.<br />
<br />
Problem B. Using the continued fraction expansion, find a solution to the Pell equation x^2 - 13 y^2 = 1.<br />
<br />
Problem C. Show that the modified Pell equation x^2 - 7y^2 = -1 has no solutions in integers x,y. (Hint: reduce the equation modulo a suitably chosen prime.)<br />
<br />
Problem D.1. Pick two values of a in F_11 = Z/11Z (a not equal to 3), such that the equation y^2 = x^3+ax+1 defines an elliptic curve (i.e., it is smooth). For each such a, determine the number of points #E(F_11) and check that it falls inside the interval described in class. <br />
<br />
Problem D.2. The point P = (0,1) lies on each of these curves. For a =3 (the curve discussed in class) determine the order of P in the elliptic curve group, that is, find the smallest positive integer n such that nP = (infinity) -- the identity element in the group law.<br />
<br />
<!-- '''November 19'''<br />
<br />
Problem A. Using the method discussed in class (which is also the method of problem 4.6 in Stein, which in retrospect I think was too hard to assign with no preparation) find a nontrivial solution with y > 0 to the equation x^2 + 11y^2 = z^2. <br />
Problem B. Let f(X) be the number of solutions of A^2 + B^2 = C^3 such that A^2, B^2, and C^3 are all at most X. Use the heuristic described in class to explain why one might expect f(X) to grow more or less like X^{1/3}.<br />
<br />
Problem C. Show that if A+Bi is the cube of a Gaussian integer, then A^2 + B^2 is a perfect cube.<br />
<br />
Do ONE of problem D1 and D2; D1 is for people who like Sage, D2 is for people who like proving things.<br />
<br />
D1. Use Sage to compute f(X) for X = 1000, 10000, 100000. Does the answer look consistent with the heuristic prediction you made in B? Does f(X)/X^{1/3} appear to be approaching a limit?<br />
D2. We will use problem C to give a lower bound for f(X). Use this to show that there are at least X^{1/3} Gaussian integers with norm at most X that are perfect cubes. From here, show that f(X) > X^{1/3}.<br />
<br />
OPTIONAL: Show that the converse of problem C also holds -- A+Bi has norm a perfect cube if and only if A+Bi is a perfect cube. (You will need unique factorization of Gaussian integers.) Using this fact, prove that f(X) is bounded above by C X^{1/3} for some constant C.<br />
<br />
'''December 8 (note nonstandard due date)'''<br />
<br />
Problem A: Hyperbolas, ellipses, and "magic slopes"<br />
<br />
In the usual analytic geometry, both hyperbolas and ellipses are given by equations of the form<br />
<br />
ax^2 + bxy + cy^2 = d (*)<br />
<br />
<br />
(we always assume d is nonzero.)<br />
<br />
How can we tell whether such an equation describes an ellipse or a<br />
hyperbola? Well, in the geometric setting we all know and love (i.e. in<br />
R^2) a hyperbola has asymptotes and an ellipse does not. What is an <br />
asymptote? You could say it's "a line which doesn't intersect the curve", <br />
but ellipses have such lines too. You might want to say "a line which <br />
doesn't intersect the curve but comes closer and closer to the curve," but <br />
the problem here is that this a) is somewhat imprecise, and b) relies on a <br />
notion of "closer" that is not going to be very clear in Z/pZ, which is <br />
our ultimate goal!<br />
<br />
<br />
So let me state it a different way: let's say the asymptotes of a <br />
hyperbola have slopes m_1 and m_2. These are "magic slopes" in the <br />
following sense: ANY line of slope m_1 (and ditto for m_2) intersects the <br />
hyperbola in at most one point. An ellipse doesn't have "magic slopes" -- <br />
you can convince yourself by drawing pictures that, for any slope m, you <br />
can find a line of slope m striking the ellipse twice.<br />
<br />
<br />
[REMARK: For technical reasons we're going to ignore hyperbolae with a <br />
vertical asymptote, since vertical lines don't exactly have slope.]<br />
<br />
<br />
Now you might know the criterion that, over the real numbers.<br />
<br />
<br />
ax^2 + bxy + cy^2 = d<br />
<br />
<br />
is a hyperbola if b^2 - 4ac > 0 and an ellipse if b^2 - 4ac < 0. (For <br />
instance, x^2 - y^2 = 1 is a hyperbola with magic slopes 1 and -1, and x^2 <br />
+ y^2 = 1 is an ellipse.) <br />
This criterion does not make sense in Z/pZ, where there is no notion of <br />
"greater" or "less."<br />
<br />
<br />
QUESTION A.1.<br />
<br />
For which primes p is it the case that <br />
<br />
x^2 + y^2 = 1<br />
<br />
has magic slopes in Z/pZ? <br />
<br />
QUESTION A.2. For which primes p is it the case that<br />
<br />
<br />
x^2 - y^2 = 1<br />
<br />
<br />
has magic slopes in Z/pZ?<br />
<br />
<br />
QUESTION B: Back in the very first week of this course we studied the set of integers x such that x^2 + 1 is prime. Based on the heuristics we discussed in class, about how many x between 1 and N would you expect to satisfy "x^2 + 1 is prime?"<br />
<br />
<br />
QUESTION C: Let z be a complex number. Then the set of all complex numbers of the form a + bz, with a and b integers, forms a lattice in the complex plane (thought of as R^2.) What is the covolume of this lattice, in terms of z?<br />
<br />
<br />
QUESTION D: Let z be a complex number. Using Minkowski's theorem and the result of question C, show that there exist integers a,b, not both zero, such that Norm(a+bz) is at most (4/pi) Im(z). (Hint: let Omega be the set of all complex numbers of norm at most (4/pi) Im(z).)<br />
<br />
<br />
OPTIONAL EXTRA: Can you give an exact formula for the minimal norm of any nonzero complex number of the form a+bz?<br />
<br />
'''December 12''' Book problems 6.1, 6.2, 6.5, 6.10. For extra credit attempt to do problem 6.9.<br />
<br />
--></div>Andreichttps://www.math.wisc.edu/wiki/index.php?title=Math_567_--_Elementary_Number_Theory&diff=19347Math 567 -- Elementary Number Theory2020-04-10T03:59:06Z<p>Andreic: </p>
<hr />
<div>'''MATH 567<br />
<br />
Elementary Number Theory'''<br />
<br />
MWF 1:20-2:10, Van Vleck B119<br />
<br />
'''Professor:''' [http://www.math.wisc.edu/~andreic/ Andrei Caldararu] (andreic@math.wisc.edu)<br />
<br />
''Office Hours:'' Wednesdays 2:30-3:30, Van Vleck 605. <br />
<br />
'''Grader:''' Yifan Peng, email: peng64@wisc.edu<br />
<br />
''Office Hours:'' TBA.<br />
<br />
Math 567 is a course in elementary number theory, aimed at undergraduates majoring in math or other quantitative disciplines. A general familiarity with abstract algebra at the level of Math 541 will be assumed, but students who haven't taken 541 are welcome to attend if they're willing to play a little catchup. We will be using William Stein's new (and cheap) textbook [http://www.amazon.com/Elementary-Number-Theory-Computational-Undergraduate/dp/0387855246 Elementary Number Theory: Primes, Congruences, and Secrets], which emphasizes computational approaches to the subject. If you don't need a physical copy of the book, [http://www.williamstein.org/ent/ it is available as a free legal .pdf.] We will be using the (free, public-domain) mathematical software [http://www.sagemath.org/ SAGE], developed largely by Stein, as an integral component of our coursework. There is a [http://doc.sagemath.org/pdf/en/tutorial/SageTutorial.pdf useful online tutorial.] You can download SAGE to your own computer or [http://www.sagenb.org use it online].<br />
<br />
Topics include some subset of, but are not limited to: Divisibility, the Euclidean algorithm and the GCD, linear Diophantine equations, prime numbers and uniqueness of factorization. Congruences, Chinese remainder theorem, Fermat's "little" theorem, Wilson's theorem, Euler's theorem and totient function, the RSA cryptosystem, Rabin's encryption scheme, Diffie-Hellman key exchange protocol. Number-theoretic functions, multiplicative functions, Möbius inversion. Primitive roots and indices. Quadratic reciprocity and the Legendre symbol. Perfect numbers, Mersenne primes, Fermat primes. Pythagorean triples, Special cases of Fermat's "last" theorem. Fibonacci numbers. Continued fractions. Distribution of primes, discussion of prime number theorem. Primality testing and factoring algorithms.<br />
<br />
'''Course Policies:''' Homework will be due on Fridays. It can be turned in late only with ''advance'' permission from your grader. It is acceptable to use calculators and computers on homework (indeed, some of it will require a computer) but calculators are not allowed during exams. You are encouraged to work together on homework, but writeups must be done individually.<br />
<br />
Many of the problems in this course will ask you to prove things. I expect proofs to be written in English sentences; the proofs in Stein's book are a good model for the level of verbosity I am looking for. <br />
<br />
'''Grading:''' The grade in Math 567 will be composed of 50% homework, 25% midterm, 25% final. The midterm will be on March 11, in class. The final exam will be on 5/8/2020, 7:45AM-9:45AM in room TBA.<br />
<br />
'''Syllabus:''' <br />
(This may change as we see what pace works well for the course. All section numbers refer to Stein's book.)<br />
<br />
* Jan 22-31: Prime numbers, prime factorizations, Euclidean algorithm and GCD (1.1-1.2)<br />
* Feb 3-7: The integers mod n, Euler's theorem, the phi function (2.1-2.2)<br />
* Feb 10-14: Modular exponentiation, primality testing, and primitive roots (2.4-2.5)<br />
* Feb 17-21: Public-key cryptography and RSA (3.1-3.4)<br />
* Feb 24-28: Rabin's algorithm (not in the book); algebraic numbers <br />
* Mar 2-6: Quadratic reciprocity (4.1-4.4)<br />
* Mar 9, Mar 13: Finite and infinite continued fractions (5.1-5.3)<br />
* Mar 11: ''Midterm exam''<br />
* Mar 23-30: Continued fractions and diophantine approximation (5.4-5.5)<br />
* Apr 1-3: Diophantine equations I: Pell's equation and Lagrange's theorem<br />
* Apr 6-10: More diophantine equations, elliptic curves (6.1)<br />
* Apr 13-17: Applications of elliptic curves (6.2-6.3)<br />
* Apr 20-24: More applications of elliptic curves (6.4)<br />
* Apr 27-May 1: advanced topic TBD: maybe additional discussion of cryptographic techniques?<br />
<br />
'''Video Lectures:'''<br />
Due to the university being transitioned to online teaching only, out lectures after spring break will need to be delivered online. I will post here videos for you to watch for the material we need to cover. Please watch them, and please give me feedback on how they could be improved.<br />
<br />
* '''Lecture for Mar. 23''': watch it [https://mediaspace.wisc.edu/media/Math+567+March+23+Lecture/0_fkhvkxxv here].<br />
* '''Lecture for Mar. 25''': watch it [https://mediaspace.wisc.edu/media/Math+567+March+25+lecture/0_qgkhpm19 here].<br />
* '''Lecture for Mar. 27''': watch it [https://mediaspace.wisc.edu/media/March+27+Math+567+Lecture+Video/0_ajbr2ja0 here].<br />
* '''Lecture for Mar. 30''': watch it [https://mediaspace.wisc.edu/media/March+30+Math+567+Lecture/0_yz44sw2c here].<br />
* '''Lecture for Apr. 1''': watch it [https://mediaspace.wisc.edu/media/April+1+Math+567/0_e64bdfcf here].<br />
* '''Lecture for Apr. 3''': watch it [https://mediaspace.wisc.edu/media/April+3+Math+567+Video/0_02rvfino here].<br />
* '''Lecture for Apr. 6''': watch it [https://mediaspace.wisc.edu/media/April+6+Math+567+Video/1_jmm8r268 here].<br />
* '''Lecture for Apr. 8''': watch it [https://mediaspace.wisc.edu/media/Group+law+elliptic/1_os81ndwu here].<br />
* '''Lecture for Apr. 10''': watch it [https://mediaspace.wisc.edu/media/Pollard%27s+p-1+method+for+factoring/1_qyf08tuv here].<br />
<br />
'''Homework:'''<br />
Homework is due at the beginning of class on the specified Friday. Typing your homework is not a requirement, but if you don't already know LaTeX I highly recommend that you learn it and use it to typeset your homework. I will sometimes assign extra problems, which I will e-mail to the class list and include here.<br />
<br />
* '''Jan 31''': 1.1, 1.3, 1.5, 1.7 (use SAGE), 1.8, 1.14.<br />
Problem A: Use SAGE to compute the number of x in [1..N] such that x^2 + 1 is prime, for N = 100, N = 1000, and N = 10000. Let f(N) be the number of such x. <br />
<br />
a) Can you formulate a conjecture about the relationship between f(N) and N/log N? <br />
<br />
b) What if x^2 + 1 is replaced with x^2 + 2? Can you explain why x^2 + 2 appears less likely to be prime? (Hint: consider x mod 3.)<br />
<br />
c) Prove that f(N) is at most (1/2)N+1. (Hint: consider x mod 2.) <br />
<br />
d) Give as good an upper bound as you can for f(N).<br />
<br />
Note that, despite the evident regularities you'll observe in this problem, we do not even know whether there are infinitely many primes of the form x^2 + 1! You would become very famous if you proved this.<br />
<br />
* '''Feb 7''': 2.6 (the formulation of numerical evidence should be done by Sage if you've got Sage working, and by calculator if not; you can use an online tool like [http://primes.utm.edu/curios/includes/primetest.php this] to test whether a number is prime.) 2.8,2.9,2.11,2.12,2.14,2.19<br />
<br />
* '''Feb 17''': 2.15, 2.18, 2.23, 2.26, 2.30.<br />
<br />
Problem A: Prove that if n=pq, with p,q prime, then n is not a Carmichael number.<br />
<br />
* '''Feb 21''': Book problems: 3.4, 3.5, 3.6<br />
<br />
Problem A. Prove that there are infinitely many primes p such that 2 is '''not''' a primitive root in Z/pZ. We break this up into steps.<br />
<br />
Problem A.1. Prove that, if x is an element of Z/nZ, then x^2 is not a primitive root.<br />
<br />
Problem A.2. Prove that there are infinitely many primes p such that 2 is a square in Z/pZ. Hint: suppose there are only finitely many such primes p_1, .. p_r, and define N = (p_1 .. p_r)^2 - 2. Where can you go from here...?<br />
<br />
Problem A.3. Give a list of five primes p such that 2 is not a primitive root in Z/pZ (you can use the method of this proof or any other.)<br />
<br />
Problem B. Prove that 24 is the largest integer n such that every element of (Z/nZ)^* is a root of x^2-1.<br />
<br />
* '''Feb 28''': Problem A. Using p = 23 and q=31, show how to encrypt the message x=240 with Rabin's algorithm. Find all possible decryptions of your encrypted message.<br />
<br />
* '''March 23''': 4.1 from the book.<br />
<br />
Problem A. Give a prime factorization of the Gaussian integer 7+9i.<br />
<br />
Problem B. Read the notes from [http://www.math.ucsd.edu/~jverstra/Gaussian1.pdf here]. Note that Z[i] satisfies a ''reduction theorem'': if n and d are Gaussian integers, then there exists integers q and r such that n = qd + r and Norm(r) < Norm(d). But (by contrast with the case of Z) this d may not be unique. In some contexts it is better to be able to choose r uniquely, even if this means letting r have norm greater than Norm(d). <br />
<br />
B.1. When d = 1+2i, show that, for each n in Z[i], there is a '''unique''' pair (q,r) in Z[i] such that n = qd+r and r is contained in the set {0,1,2,3,4}. For instance, i can be written as i(1+2i) + 2, so we say i reduces to 2 mod (1+2i).<br />
(Hint: Suppose q(1+2i)+r = q'(1+2i) + r'. What can we say about (r-r'), and why is this incompatible with both r and r' being in {0,1,2,3,4}?<br />
<br />
B.2. Show that if n is an integer in Z, the reduction of n mod (1+2i) is equal to its reduction mod 5.<br />
<br />
Problem C. Let's try to figure out how to define "phi(d)" for a Gaussian integer d. Suppose S is a set of Gaussian integers such that every n in Z[i] can be written uniquely as qd+r, with q in Z[i] and r in S. (So for instance when d=1+2i, we showed in problem B that we can take S to be {0...4}. It would also be OK to take S to be {1..5} or {0,i,2i,1+i,1+2i}. In fact, it turns out that S has to be a set of size Norm(d) (I might or might not prove this in class; if not, feel free just to accept it.)<br />
<br />
Now define phi(d) to be the number of elements s of S such that s and d are coprime.<br />
<br />
C.1. Compute phi(1+2i) and phi(3).<br />
<br />
C.2. Prove that the value of phi(d) does not depend on the choice of S.<br />
<br />
C.3. Prove that every n in Z[i] which is coprime to 3 satisfies n^phi(3) = 1 mod 3; that is, Euler's theorem holds in this case. (You can prove this by direct computation; of course, if you want, you are welcome to prove that Euler's theorem holds for Z[i] in general, adapting the proof in Stein's book or the one we gave in class.)<br />
<br />
*'''March 30''' <br />
<br />
Problem A. Express 50005 as the sum of two squares.<br />
<br />
In the next two problems we denote by r(n) the number of ways to express n as the sum of two squares (i.e. the number of ordered pairs (a,b) such that a^2 + b^2 = n.) For instance, r(5) = 8: (a,b) = (+/-1, +/-2) or (+/-2, +/-1).<br />
<br />
Problem B. Prove that, for any N, there exists an integer n such that r(n) > N. (I.E., the function r(n) is ''unbounded.'')<br />
<br />
Problem C. If you like Sage, write a short program in Sage to compute r(n) and compute the average of r(n) as n ranges from 1 to 1000. Whether or not you like Sage, make a guess as to how this average would behave if you replaced "1000" by a larger and larger number. (Feel free to ask the Sage-lovers what answer they got in the optional first part of the question.) Can you prove this guess is correct?<br />
<br />
Problem D. We saw in class that the ring Z[sqrt(-5)] doesn't have unique factorization; 6 can be factored as 2*3 or (1+sqrt(-5))(1-sqrt(-5)). In this problem, we will prove that Z[sqrt(-d)] fails to have unique factorization for EVERY odd d >= 5. (Actually it's true for all d >= 5 but to make the proof manageable we'll restrict to the odd case.]<br />
<br />
D.1. Show that (1+sqrt(-d)), (1-sqrt(-d)) and 2 are irreducible in Z[sqrt(-d)].<br />
<br />
D.2. Now give an element of Z[sqrt(-d)] that has two distinct factorizations into irreducibles. (Hint: imitate the example we used for Z[sqrt(-5)].)<br />
''Remark:'' The rings Z[sqrt(d)], where d is positive, are quite different -- here we believe that there are infinitely many with unique factorization, though this conjecture has remained unproved for many decades!<br />
<br />
<!--<br />
*'''November 16'''<br />
<br />
Book problems: 5.3,5.4,5.5<br />
<br />
Problem A: We discussed in class the problem of studying which positive integers are the sum of two squares. In this problem we prove that every element n of (Z/pZ) is the sum of two squares. We argue as follows. Let S be the set of squares in (Z/pZ), and let T be the set of (Z/pZ) consisting of all elements of the form n - x^2, for some x in (Z/pZ).<br />
<br />
A.1. What is the size of S and of T? Use this to show that S and T are not disjoint.<br />
<br />
A.2. Given that S and T are not disjoint, prove that n is the sum of two squares.<br />
<br />
Problem B. Using the continued fraction expansion, find a solution to the Pell equation x^2 - 13 y^2 = 1.<br />
<br />
Problem C. Show that the modified Pell equation x^2 - 7y^2 = -1 has no solutions in integers x,y. (Hint: reduce the equation modulo a suitably chosen prime.)<br />
<br />
Problem D. Stuffy Stirnweiss finished the 1945 season with a batting average of .3085443. Using continued fractions, guess how many at-bats he had. Tony Cuccinello had a batting average of .3084577. Given that he had more than 200 and fewer than 600 at-bats, can you estimate the number of at-bats he had?<br />
<br />
'''December 7'''<br />
<br />
Problem A. When p is a prime congruent to 3 mod 4, prove that ((p-1)/2)! is either 1 or -1 in (Z/pZ). '''OPTIONAL:''' Use sage to compute whether this factorial is 1 or -1 for many primes p. Is there any pattern? Does it seem to be 1 half the time and -1 half the time? Note that I have no idea what the answer to this question is.<br />
<br />
Problem B. Using the continued fraction expansion, find a solution to the Pell equation x^2 - 13 y^2 = 1.<br />
<br />
Problem C. Show that the modified Pell equation x^2 - 7y^2 = -1 has no solutions in integers x,y. (Hint: reduce the equation modulo a suitably chosen prime.)<br />
<br />
Problem D.1. Pick two values of a in F_11 = Z/11Z (a not equal to 3), such that the equation y^2 = x^3+ax+1 defines an elliptic curve (i.e., it is smooth). For each such a, determine the number of points #E(F_11) and check that it falls inside the interval described in class. <br />
<br />
Problem D.2. The point P = (0,1) lies on each of these curves. For a =3 (the curve discussed in class) determine the order of P in the elliptic curve group, that is, find the smallest positive integer n such that nP = (infinity) -- the identity element in the group law.<br />
<br />
<!-- '''November 19'''<br />
<br />
Problem A. Using the method discussed in class (which is also the method of problem 4.6 in Stein, which in retrospect I think was too hard to assign with no preparation) find a nontrivial solution with y > 0 to the equation x^2 + 11y^2 = z^2. <br />
Problem B. Let f(X) be the number of solutions of A^2 + B^2 = C^3 such that A^2, B^2, and C^3 are all at most X. Use the heuristic described in class to explain why one might expect f(X) to grow more or less like X^{1/3}.<br />
<br />
Problem C. Show that if A+Bi is the cube of a Gaussian integer, then A^2 + B^2 is a perfect cube.<br />
<br />
Do ONE of problem D1 and D2; D1 is for people who like Sage, D2 is for people who like proving things.<br />
<br />
D1. Use Sage to compute f(X) for X = 1000, 10000, 100000. Does the answer look consistent with the heuristic prediction you made in B? Does f(X)/X^{1/3} appear to be approaching a limit?<br />
D2. We will use problem C to give a lower bound for f(X). Use this to show that there are at least X^{1/3} Gaussian integers with norm at most X that are perfect cubes. From here, show that f(X) > X^{1/3}.<br />
<br />
OPTIONAL: Show that the converse of problem C also holds -- A+Bi has norm a perfect cube if and only if A+Bi is a perfect cube. (You will need unique factorization of Gaussian integers.) Using this fact, prove that f(X) is bounded above by C X^{1/3} for some constant C.<br />
<br />
'''December 8 (note nonstandard due date)'''<br />
<br />
Problem A: Hyperbolas, ellipses, and "magic slopes"<br />
<br />
In the usual analytic geometry, both hyperbolas and ellipses are given by equations of the form<br />
<br />
ax^2 + bxy + cy^2 = d (*)<br />
<br />
<br />
(we always assume d is nonzero.)<br />
<br />
How can we tell whether such an equation describes an ellipse or a<br />
hyperbola? Well, in the geometric setting we all know and love (i.e. in<br />
R^2) a hyperbola has asymptotes and an ellipse does not. What is an <br />
asymptote? You could say it's "a line which doesn't intersect the curve", <br />
but ellipses have such lines too. You might want to say "a line which <br />
doesn't intersect the curve but comes closer and closer to the curve," but <br />
the problem here is that this a) is somewhat imprecise, and b) relies on a <br />
notion of "closer" that is not going to be very clear in Z/pZ, which is <br />
our ultimate goal!<br />
<br />
<br />
So let me state it a different way: let's say the asymptotes of a <br />
hyperbola have slopes m_1 and m_2. These are "magic slopes" in the <br />
following sense: ANY line of slope m_1 (and ditto for m_2) intersects the <br />
hyperbola in at most one point. An ellipse doesn't have "magic slopes" -- <br />
you can convince yourself by drawing pictures that, for any slope m, you <br />
can find a line of slope m striking the ellipse twice.<br />
<br />
<br />
[REMARK: For technical reasons we're going to ignore hyperbolae with a <br />
vertical asymptote, since vertical lines don't exactly have slope.]<br />
<br />
<br />
Now you might know the criterion that, over the real numbers.<br />
<br />
<br />
ax^2 + bxy + cy^2 = d<br />
<br />
<br />
is a hyperbola if b^2 - 4ac > 0 and an ellipse if b^2 - 4ac < 0. (For <br />
instance, x^2 - y^2 = 1 is a hyperbola with magic slopes 1 and -1, and x^2 <br />
+ y^2 = 1 is an ellipse.) <br />
This criterion does not make sense in Z/pZ, where there is no notion of <br />
"greater" or "less."<br />
<br />
<br />
QUESTION A.1.<br />
<br />
For which primes p is it the case that <br />
<br />
x^2 + y^2 = 1<br />
<br />
has magic slopes in Z/pZ? <br />
<br />
QUESTION A.2. For which primes p is it the case that<br />
<br />
<br />
x^2 - y^2 = 1<br />
<br />
<br />
has magic slopes in Z/pZ?<br />
<br />
<br />
QUESTION B: Back in the very first week of this course we studied the set of integers x such that x^2 + 1 is prime. Based on the heuristics we discussed in class, about how many x between 1 and N would you expect to satisfy "x^2 + 1 is prime?"<br />
<br />
<br />
QUESTION C: Let z be a complex number. Then the set of all complex numbers of the form a + bz, with a and b integers, forms a lattice in the complex plane (thought of as R^2.) What is the covolume of this lattice, in terms of z?<br />
<br />
<br />
QUESTION D: Let z be a complex number. Using Minkowski's theorem and the result of question C, show that there exist integers a,b, not both zero, such that Norm(a+bz) is at most (4/pi) Im(z). (Hint: let Omega be the set of all complex numbers of norm at most (4/pi) Im(z).)<br />
<br />
<br />
OPTIONAL EXTRA: Can you give an exact formula for the minimal norm of any nonzero complex number of the form a+bz?<br />
<br />
'''December 12''' Book problems 6.1, 6.2, 6.5, 6.10. For extra credit attempt to do problem 6.9.<br />
<br />
--></div>Andreichttps://www.math.wisc.edu/wiki/index.php?title=Math_567_--_Elementary_Number_Theory&diff=19344Math 567 -- Elementary Number Theory2020-04-08T04:04:52Z<p>Andreic: </p>
<hr />
<div>'''MATH 567<br />
<br />
Elementary Number Theory'''<br />
<br />
MWF 1:20-2:10, Van Vleck B119<br />
<br />
'''Professor:''' [http://www.math.wisc.edu/~andreic/ Andrei Caldararu] (andreic@math.wisc.edu)<br />
<br />
''Office Hours:'' Wednesdays 2:30-3:30, Van Vleck 605. <br />
<br />
'''Grader:''' Yifan Peng, email: peng64@wisc.edu<br />
<br />
''Office Hours:'' TBA.<br />
<br />
Math 567 is a course in elementary number theory, aimed at undergraduates majoring in math or other quantitative disciplines. A general familiarity with abstract algebra at the level of Math 541 will be assumed, but students who haven't taken 541 are welcome to attend if they're willing to play a little catchup. We will be using William Stein's new (and cheap) textbook [http://www.amazon.com/Elementary-Number-Theory-Computational-Undergraduate/dp/0387855246 Elementary Number Theory: Primes, Congruences, and Secrets], which emphasizes computational approaches to the subject. If you don't need a physical copy of the book, [http://www.williamstein.org/ent/ it is available as a free legal .pdf.] We will be using the (free, public-domain) mathematical software [http://www.sagemath.org/ SAGE], developed largely by Stein, as an integral component of our coursework. There is a [http://doc.sagemath.org/pdf/en/tutorial/SageTutorial.pdf useful online tutorial.] You can download SAGE to your own computer or [http://www.sagenb.org use it online].<br />
<br />
Topics include some subset of, but are not limited to: Divisibility, the Euclidean algorithm and the GCD, linear Diophantine equations, prime numbers and uniqueness of factorization. Congruences, Chinese remainder theorem, Fermat's "little" theorem, Wilson's theorem, Euler's theorem and totient function, the RSA cryptosystem, Rabin's encryption scheme, Diffie-Hellman key exchange protocol. Number-theoretic functions, multiplicative functions, Möbius inversion. Primitive roots and indices. Quadratic reciprocity and the Legendre symbol. Perfect numbers, Mersenne primes, Fermat primes. Pythagorean triples, Special cases of Fermat's "last" theorem. Fibonacci numbers. Continued fractions. Distribution of primes, discussion of prime number theorem. Primality testing and factoring algorithms.<br />
<br />
'''Course Policies:''' Homework will be due on Fridays. It can be turned in late only with ''advance'' permission from your grader. It is acceptable to use calculators and computers on homework (indeed, some of it will require a computer) but calculators are not allowed during exams. You are encouraged to work together on homework, but writeups must be done individually.<br />
<br />
Many of the problems in this course will ask you to prove things. I expect proofs to be written in English sentences; the proofs in Stein's book are a good model for the level of verbosity I am looking for. <br />
<br />
'''Grading:''' The grade in Math 567 will be composed of 50% homework, 25% midterm, 25% final. The midterm will be on March 11, in class. The final exam will be on 5/8/2020, 7:45AM-9:45AM in room TBA.<br />
<br />
'''Syllabus:''' <br />
(This may change as we see what pace works well for the course. All section numbers refer to Stein's book.)<br />
<br />
* Jan 22-31: Prime numbers, prime factorizations, Euclidean algorithm and GCD (1.1-1.2)<br />
* Feb 3-7: The integers mod n, Euler's theorem, the phi function (2.1-2.2)<br />
* Feb 10-14: Modular exponentiation, primality testing, and primitive roots (2.4-2.5)<br />
* Feb 17-21: Public-key cryptography and RSA (3.1-3.4)<br />
* Feb 24-28: Rabin's algorithm (not in the book); algebraic numbers <br />
* Mar 2-6: Quadratic reciprocity (4.1-4.4)<br />
* Mar 9, Mar 13: Finite and infinite continued fractions (5.1-5.3)<br />
* Mar 11: ''Midterm exam''<br />
* Mar 23-30: Continued fractions and diophantine approximation (5.4-5.5)<br />
* Apr 1-3: Diophantine equations I: Pell's equation and Lagrange's theorem<br />
* Apr 6-10: More diophantine equations, elliptic curves (6.1)<br />
* Apr 13-17: Applications of elliptic curves (6.2-6.3)<br />
* Apr 20-24: More applications of elliptic curves (6.4)<br />
* Apr 27-May 1: advanced topic TBD: maybe additional discussion of cryptographic techniques?<br />
<br />
'''Video Lectures:'''<br />
Due to the university being transitioned to online teaching only, out lectures after spring break will need to be delivered online. I will post here videos for you to watch for the material we need to cover. Please watch them, and please give me feedback on how they could be improved.<br />
<br />
* '''Lecture for Mar. 23''': watch it [https://mediaspace.wisc.edu/media/Math+567+March+23+Lecture/0_fkhvkxxv here].<br />
* '''Lecture for Mar. 25''': watch it [https://mediaspace.wisc.edu/media/Math+567+March+25+lecture/0_qgkhpm19 here].<br />
* '''Lecture for Mar. 27''': watch it [https://mediaspace.wisc.edu/media/March+27+Math+567+Lecture+Video/0_ajbr2ja0 here].<br />
* '''Lecture for Mar. 30''': watch it [https://mediaspace.wisc.edu/media/March+30+Math+567+Lecture/0_yz44sw2c here].<br />
* '''Lecture for Apr. 1''': watch it [https://mediaspace.wisc.edu/media/April+1+Math+567/0_e64bdfcf here].<br />
* '''Lecture for Apr. 3''': watch it [https://mediaspace.wisc.edu/media/April+3+Math+567+Video/0_02rvfino here].<br />
* '''Lecture for Apr. 6''': watch it [https://mediaspace.wisc.edu/media/April+6+Math+567+Video/1_jmm8r268 here].<br />
* '''Lecture for Apr. 8''': watch it [https://mediaspace.wisc.edu/media/Group+law+elliptic/1_os81ndwu here].<br />
<br />
'''Homework:'''<br />
Homework is due at the beginning of class on the specified Friday. Typing your homework is not a requirement, but if you don't already know LaTeX I highly recommend that you learn it and use it to typeset your homework. I will sometimes assign extra problems, which I will e-mail to the class list and include here.<br />
<br />
* '''Jan 31''': 1.1, 1.3, 1.5, 1.7 (use SAGE), 1.8, 1.14.<br />
Problem A: Use SAGE to compute the number of x in [1..N] such that x^2 + 1 is prime, for N = 100, N = 1000, and N = 10000. Let f(N) be the number of such x. <br />
<br />
a) Can you formulate a conjecture about the relationship between f(N) and N/log N? <br />
<br />
b) What if x^2 + 1 is replaced with x^2 + 2? Can you explain why x^2 + 2 appears less likely to be prime? (Hint: consider x mod 3.)<br />
<br />
c) Prove that f(N) is at most (1/2)N+1. (Hint: consider x mod 2.) <br />
<br />
d) Give as good an upper bound as you can for f(N).<br />
<br />
Note that, despite the evident regularities you'll observe in this problem, we do not even know whether there are infinitely many primes of the form x^2 + 1! You would become very famous if you proved this.<br />
<br />
* '''Feb 7''': 2.6 (the formulation of numerical evidence should be done by Sage if you've got Sage working, and by calculator if not; you can use an online tool like [http://primes.utm.edu/curios/includes/primetest.php this] to test whether a number is prime.) 2.8,2.9,2.11,2.12,2.14,2.19<br />
<br />
* '''Feb 17''': 2.15, 2.18, 2.23, 2.26, 2.30.<br />
<br />
Problem A: Prove that if n=pq, with p,q prime, then n is not a Carmichael number.<br />
<br />
* '''Feb 21''': Book problems: 3.4, 3.5, 3.6<br />
<br />
Problem A. Prove that there are infinitely many primes p such that 2 is '''not''' a primitive root in Z/pZ. We break this up into steps.<br />
<br />
Problem A.1. Prove that, if x is an element of Z/nZ, then x^2 is not a primitive root.<br />
<br />
Problem A.2. Prove that there are infinitely many primes p such that 2 is a square in Z/pZ. Hint: suppose there are only finitely many such primes p_1, .. p_r, and define N = (p_1 .. p_r)^2 - 2. Where can you go from here...?<br />
<br />
Problem A.3. Give a list of five primes p such that 2 is not a primitive root in Z/pZ (you can use the method of this proof or any other.)<br />
<br />
Problem B. Prove that 24 is the largest integer n such that every element of (Z/nZ)^* is a root of x^2-1.<br />
<br />
* '''Feb 28''': Problem A. Using p = 23 and q=31, show how to encrypt the message x=240 with Rabin's algorithm. Find all possible decryptions of your encrypted message.<br />
<br />
* '''March 23''': 4.1 from the book.<br />
<br />
Problem A. Give a prime factorization of the Gaussian integer 7+9i.<br />
<br />
Problem B. Read the notes from [http://www.math.ucsd.edu/~jverstra/Gaussian1.pdf here]. Note that Z[i] satisfies a ''reduction theorem'': if n and d are Gaussian integers, then there exists integers q and r such that n = qd + r and Norm(r) < Norm(d). But (by contrast with the case of Z) this d may not be unique. In some contexts it is better to be able to choose r uniquely, even if this means letting r have norm greater than Norm(d). <br />
<br />
B.1. When d = 1+2i, show that, for each n in Z[i], there is a '''unique''' pair (q,r) in Z[i] such that n = qd+r and r is contained in the set {0,1,2,3,4}. For instance, i can be written as i(1+2i) + 2, so we say i reduces to 2 mod (1+2i).<br />
(Hint: Suppose q(1+2i)+r = q'(1+2i) + r'. What can we say about (r-r'), and why is this incompatible with both r and r' being in {0,1,2,3,4}?<br />
<br />
B.2. Show that if n is an integer in Z, the reduction of n mod (1+2i) is equal to its reduction mod 5.<br />
<br />
Problem C. Let's try to figure out how to define "phi(d)" for a Gaussian integer d. Suppose S is a set of Gaussian integers such that every n in Z[i] can be written uniquely as qd+r, with q in Z[i] and r in S. (So for instance when d=1+2i, we showed in problem B that we can take S to be {0...4}. It would also be OK to take S to be {1..5} or {0,i,2i,1+i,1+2i}. In fact, it turns out that S has to be a set of size Norm(d) (I might or might not prove this in class; if not, feel free just to accept it.)<br />
<br />
Now define phi(d) to be the number of elements s of S such that s and d are coprime.<br />
<br />
C.1. Compute phi(1+2i) and phi(3).<br />
<br />
C.2. Prove that the value of phi(d) does not depend on the choice of S.<br />
<br />
C.3. Prove that every n in Z[i] which is coprime to 3 satisfies n^phi(3) = 1 mod 3; that is, Euler's theorem holds in this case. (You can prove this by direct computation; of course, if you want, you are welcome to prove that Euler's theorem holds for Z[i] in general, adapting the proof in Stein's book or the one we gave in class.)<br />
<br />
*'''March 30''' <br />
<br />
Problem A. Express 50005 as the sum of two squares.<br />
<br />
In the next two problems we denote by r(n) the number of ways to express n as the sum of two squares (i.e. the number of ordered pairs (a,b) such that a^2 + b^2 = n.) For instance, r(5) = 8: (a,b) = (+/-1, +/-2) or (+/-2, +/-1).<br />
<br />
Problem B. Prove that, for any N, there exists an integer n such that r(n) > N. (I.E., the function r(n) is ''unbounded.'')<br />
<br />
Problem C. If you like Sage, write a short program in Sage to compute r(n) and compute the average of r(n) as n ranges from 1 to 1000. Whether or not you like Sage, make a guess as to how this average would behave if you replaced "1000" by a larger and larger number. (Feel free to ask the Sage-lovers what answer they got in the optional first part of the question.) Can you prove this guess is correct?<br />
<br />
Problem D. We saw in class that the ring Z[sqrt(-5)] doesn't have unique factorization; 6 can be factored as 2*3 or (1+sqrt(-5))(1-sqrt(-5)). In this problem, we will prove that Z[sqrt(-d)] fails to have unique factorization for EVERY odd d >= 5. (Actually it's true for all d >= 5 but to make the proof manageable we'll restrict to the odd case.]<br />
<br />
D.1. Show that (1+sqrt(-d)), (1-sqrt(-d)) and 2 are irreducible in Z[sqrt(-d)].<br />
<br />
D.2. Now give an element of Z[sqrt(-d)] that has two distinct factorizations into irreducibles. (Hint: imitate the example we used for Z[sqrt(-5)].)<br />
''Remark:'' The rings Z[sqrt(d)], where d is positive, are quite different -- here we believe that there are infinitely many with unique factorization, though this conjecture has remained unproved for many decades!<br />
<br />
<!--<br />
*'''November 16'''<br />
<br />
Book problems: 5.3,5.4,5.5<br />
<br />
Problem A: We discussed in class the problem of studying which positive integers are the sum of two squares. In this problem we prove that every element n of (Z/pZ) is the sum of two squares. We argue as follows. Let S be the set of squares in (Z/pZ), and let T be the set of (Z/pZ) consisting of all elements of the form n - x^2, for some x in (Z/pZ).<br />
<br />
A.1. What is the size of S and of T? Use this to show that S and T are not disjoint.<br />
<br />
A.2. Given that S and T are not disjoint, prove that n is the sum of two squares.<br />
<br />
Problem B. Using the continued fraction expansion, find a solution to the Pell equation x^2 - 13 y^2 = 1.<br />
<br />
Problem C. Show that the modified Pell equation x^2 - 7y^2 = -1 has no solutions in integers x,y. (Hint: reduce the equation modulo a suitably chosen prime.)<br />
<br />
Problem D. Stuffy Stirnweiss finished the 1945 season with a batting average of .3085443. Using continued fractions, guess how many at-bats he had. Tony Cuccinello had a batting average of .3084577. Given that he had more than 200 and fewer than 600 at-bats, can you estimate the number of at-bats he had?<br />
<br />
'''December 7'''<br />
<br />
Problem A. When p is a prime congruent to 3 mod 4, prove that ((p-1)/2)! is either 1 or -1 in (Z/pZ). '''OPTIONAL:''' Use sage to compute whether this factorial is 1 or -1 for many primes p. Is there any pattern? Does it seem to be 1 half the time and -1 half the time? Note that I have no idea what the answer to this question is.<br />
<br />
Problem B. Using the continued fraction expansion, find a solution to the Pell equation x^2 - 13 y^2 = 1.<br />
<br />
Problem C. Show that the modified Pell equation x^2 - 7y^2 = -1 has no solutions in integers x,y. (Hint: reduce the equation modulo a suitably chosen prime.)<br />
<br />
Problem D.1. Pick two values of a in F_11 = Z/11Z (a not equal to 3), such that the equation y^2 = x^3+ax+1 defines an elliptic curve (i.e., it is smooth). For each such a, determine the number of points #E(F_11) and check that it falls inside the interval described in class. <br />
<br />
Problem D.2. The point P = (0,1) lies on each of these curves. For a =3 (the curve discussed in class) determine the order of P in the elliptic curve group, that is, find the smallest positive integer n such that nP = (infinity) -- the identity element in the group law.<br />
<br />
<!-- '''November 19'''<br />
<br />
Problem A. Using the method discussed in class (which is also the method of problem 4.6 in Stein, which in retrospect I think was too hard to assign with no preparation) find a nontrivial solution with y > 0 to the equation x^2 + 11y^2 = z^2. <br />
Problem B. Let f(X) be the number of solutions of A^2 + B^2 = C^3 such that A^2, B^2, and C^3 are all at most X. Use the heuristic described in class to explain why one might expect f(X) to grow more or less like X^{1/3}.<br />
<br />
Problem C. Show that if A+Bi is the cube of a Gaussian integer, then A^2 + B^2 is a perfect cube.<br />
<br />
Do ONE of problem D1 and D2; D1 is for people who like Sage, D2 is for people who like proving things.<br />
<br />
D1. Use Sage to compute f(X) for X = 1000, 10000, 100000. Does the answer look consistent with the heuristic prediction you made in B? Does f(X)/X^{1/3} appear to be approaching a limit?<br />
D2. We will use problem C to give a lower bound for f(X). Use this to show that there are at least X^{1/3} Gaussian integers with norm at most X that are perfect cubes. From here, show that f(X) > X^{1/3}.<br />
<br />
OPTIONAL: Show that the converse of problem C also holds -- A+Bi has norm a perfect cube if and only if A+Bi is a perfect cube. (You will need unique factorization of Gaussian integers.) Using this fact, prove that f(X) is bounded above by C X^{1/3} for some constant C.<br />
<br />
'''December 8 (note nonstandard due date)'''<br />
<br />
Problem A: Hyperbolas, ellipses, and "magic slopes"<br />
<br />
In the usual analytic geometry, both hyperbolas and ellipses are given by equations of the form<br />
<br />
ax^2 + bxy + cy^2 = d (*)<br />
<br />
<br />
(we always assume d is nonzero.)<br />
<br />
How can we tell whether such an equation describes an ellipse or a<br />
hyperbola? Well, in the geometric setting we all know and love (i.e. in<br />
R^2) a hyperbola has asymptotes and an ellipse does not. What is an <br />
asymptote? You could say it's "a line which doesn't intersect the curve", <br />
but ellipses have such lines too. You might want to say "a line which <br />
doesn't intersect the curve but comes closer and closer to the curve," but <br />
the problem here is that this a) is somewhat imprecise, and b) relies on a <br />
notion of "closer" that is not going to be very clear in Z/pZ, which is <br />
our ultimate goal!<br />
<br />
<br />
So let me state it a different way: let's say the asymptotes of a <br />
hyperbola have slopes m_1 and m_2. These are "magic slopes" in the <br />
following sense: ANY line of slope m_1 (and ditto for m_2) intersects the <br />
hyperbola in at most one point. An ellipse doesn't have "magic slopes" -- <br />
you can convince yourself by drawing pictures that, for any slope m, you <br />
can find a line of slope m striking the ellipse twice.<br />
<br />
<br />
[REMARK: For technical reasons we're going to ignore hyperbolae with a <br />
vertical asymptote, since vertical lines don't exactly have slope.]<br />
<br />
<br />
Now you might know the criterion that, over the real numbers.<br />
<br />
<br />
ax^2 + bxy + cy^2 = d<br />
<br />
<br />
is a hyperbola if b^2 - 4ac > 0 and an ellipse if b^2 - 4ac < 0. (For <br />
instance, x^2 - y^2 = 1 is a hyperbola with magic slopes 1 and -1, and x^2 <br />
+ y^2 = 1 is an ellipse.) <br />
This criterion does not make sense in Z/pZ, where there is no notion of <br />
"greater" or "less."<br />
<br />
<br />
QUESTION A.1.<br />
<br />
For which primes p is it the case that <br />
<br />
x^2 + y^2 = 1<br />
<br />
has magic slopes in Z/pZ? <br />
<br />
QUESTION A.2. For which primes p is it the case that<br />
<br />
<br />
x^2 - y^2 = 1<br />
<br />
<br />
has magic slopes in Z/pZ?<br />
<br />
<br />
QUESTION B: Back in the very first week of this course we studied the set of integers x such that x^2 + 1 is prime. Based on the heuristics we discussed in class, about how many x between 1 and N would you expect to satisfy "x^2 + 1 is prime?"<br />
<br />
<br />
QUESTION C: Let z be a complex number. Then the set of all complex numbers of the form a + bz, with a and b integers, forms a lattice in the complex plane (thought of as R^2.) What is the covolume of this lattice, in terms of z?<br />
<br />
<br />
QUESTION D: Let z be a complex number. Using Minkowski's theorem and the result of question C, show that there exist integers a,b, not both zero, such that Norm(a+bz) is at most (4/pi) Im(z). (Hint: let Omega be the set of all complex numbers of norm at most (4/pi) Im(z).)<br />
<br />
<br />
OPTIONAL EXTRA: Can you give an exact formula for the minimal norm of any nonzero complex number of the form a+bz?<br />
<br />
'''December 12''' Book problems 6.1, 6.2, 6.5, 6.10. For extra credit attempt to do problem 6.9.<br />
<br />
--></div>Andreichttps://www.math.wisc.edu/wiki/index.php?title=Math_567_--_Elementary_Number_Theory&diff=19331Math 567 -- Elementary Number Theory2020-04-06T03:34:44Z<p>Andreic: </p>
<hr />
<div>'''MATH 567<br />
<br />
Elementary Number Theory'''<br />
<br />
MWF 1:20-2:10, Van Vleck B119<br />
<br />
'''Professor:''' [http://www.math.wisc.edu/~andreic/ Andrei Caldararu] (andreic@math.wisc.edu)<br />
<br />
''Office Hours:'' Wednesdays 2:30-3:30, Van Vleck 605. <br />
<br />
'''Grader:''' Yifan Peng, email: peng64@wisc.edu<br />
<br />
''Office Hours:'' TBA.<br />
<br />
Math 567 is a course in elementary number theory, aimed at undergraduates majoring in math or other quantitative disciplines. A general familiarity with abstract algebra at the level of Math 541 will be assumed, but students who haven't taken 541 are welcome to attend if they're willing to play a little catchup. We will be using William Stein's new (and cheap) textbook [http://www.amazon.com/Elementary-Number-Theory-Computational-Undergraduate/dp/0387855246 Elementary Number Theory: Primes, Congruences, and Secrets], which emphasizes computational approaches to the subject. If you don't need a physical copy of the book, [http://www.williamstein.org/ent/ it is available as a free legal .pdf.] We will be using the (free, public-domain) mathematical software [http://www.sagemath.org/ SAGE], developed largely by Stein, as an integral component of our coursework. There is a [http://doc.sagemath.org/pdf/en/tutorial/SageTutorial.pdf useful online tutorial.] You can download SAGE to your own computer or [http://www.sagenb.org use it online].<br />
<br />
Topics include some subset of, but are not limited to: Divisibility, the Euclidean algorithm and the GCD, linear Diophantine equations, prime numbers and uniqueness of factorization. Congruences, Chinese remainder theorem, Fermat's "little" theorem, Wilson's theorem, Euler's theorem and totient function, the RSA cryptosystem, Rabin's encryption scheme, Diffie-Hellman key exchange protocol. Number-theoretic functions, multiplicative functions, Möbius inversion. Primitive roots and indices. Quadratic reciprocity and the Legendre symbol. Perfect numbers, Mersenne primes, Fermat primes. Pythagorean triples, Special cases of Fermat's "last" theorem. Fibonacci numbers. Continued fractions. Distribution of primes, discussion of prime number theorem. Primality testing and factoring algorithms.<br />
<br />
'''Course Policies:''' Homework will be due on Fridays. It can be turned in late only with ''advance'' permission from your grader. It is acceptable to use calculators and computers on homework (indeed, some of it will require a computer) but calculators are not allowed during exams. You are encouraged to work together on homework, but writeups must be done individually.<br />
<br />
Many of the problems in this course will ask you to prove things. I expect proofs to be written in English sentences; the proofs in Stein's book are a good model for the level of verbosity I am looking for. <br />
<br />
'''Grading:''' The grade in Math 567 will be composed of 50% homework, 25% midterm, 25% final. The midterm will be on March 11, in class. The final exam will be on 5/8/2020, 7:45AM-9:45AM in room TBA.<br />
<br />
'''Syllabus:''' <br />
(This may change as we see what pace works well for the course. All section numbers refer to Stein's book.)<br />
<br />
* Jan 22-31: Prime numbers, prime factorizations, Euclidean algorithm and GCD (1.1-1.2)<br />
* Feb 3-7: The integers mod n, Euler's theorem, the phi function (2.1-2.2)<br />
* Feb 10-14: Modular exponentiation, primality testing, and primitive roots (2.4-2.5)<br />
* Feb 17-21: Public-key cryptography and RSA (3.1-3.4)<br />
* Feb 24-28: Rabin's algorithm (not in the book); algebraic numbers <br />
* Mar 2-6: Quadratic reciprocity (4.1-4.4)<br />
* Mar 9, Mar 13: Finite and infinite continued fractions (5.1-5.3)<br />
* Mar 11: ''Midterm exam''<br />
* Mar 23-30: Continued fractions and diophantine approximation (5.4-5.5)<br />
* Apr 1-3: Diophantine equations I: Pell's equation and Lagrange's theorem<br />
* Apr 6-10: More diophantine equations, elliptic curves (6.1)<br />
* Apr 13-17: Applications of elliptic curves (6.2-6.3)<br />
* Apr 20-24: More applications of elliptic curves (6.4)<br />
* Apr 27-May 1: advanced topic TBD: maybe additional discussion of cryptographic techniques?<br />
<br />
'''Video Lectures:'''<br />
Due to the university being transitioned to online teaching only, out lectures after spring break will need to be delivered online. I will post here videos for you to watch for the material we need to cover. Please watch them, and please give me feedback on how they could be improved.<br />
<br />
* '''Lecture for Mar. 23''': watch it [https://mediaspace.wisc.edu/media/Math+567+March+23+Lecture/0_fkhvkxxv here].<br />
* '''Lecture for Mar. 25''': watch it [https://mediaspace.wisc.edu/media/Math+567+March+25+lecture/0_qgkhpm19 here].<br />
* '''Lecture for Mar. 27''': watch it [https://mediaspace.wisc.edu/media/March+27+Math+567+Lecture+Video/0_ajbr2ja0 here].<br />
* '''Lecture for Mar. 30''': watch it [https://mediaspace.wisc.edu/media/March+30+Math+567+Lecture/0_yz44sw2c here].<br />
* '''Lecture for Apr. 1''': watch it [https://mediaspace.wisc.edu/media/April+1+Math+567/0_e64bdfcf here].<br />
* '''Lecture for Apr. 3''': watch it [https://mediaspace.wisc.edu/media/April+3+Math+567+Video/0_02rvfino here].<br />
* '''Lecture for Apr. 6''': watch it [https://mediaspace.wisc.edu/media/April+6+Math+567+Video/1_jmm8r268 here].<br />
<br />
'''Homework:'''<br />
Homework is due at the beginning of class on the specified Friday. Typing your homework is not a requirement, but if you don't already know LaTeX I highly recommend that you learn it and use it to typeset your homework. I will sometimes assign extra problems, which I will e-mail to the class list and include here.<br />
<br />
* '''Jan 31''': 1.1, 1.3, 1.5, 1.7 (use SAGE), 1.8, 1.14.<br />
Problem A: Use SAGE to compute the number of x in [1..N] such that x^2 + 1 is prime, for N = 100, N = 1000, and N = 10000. Let f(N) be the number of such x. <br />
<br />
a) Can you formulate a conjecture about the relationship between f(N) and N/log N? <br />
<br />
b) What if x^2 + 1 is replaced with x^2 + 2? Can you explain why x^2 + 2 appears less likely to be prime? (Hint: consider x mod 3.)<br />
<br />
c) Prove that f(N) is at most (1/2)N+1. (Hint: consider x mod 2.) <br />
<br />
d) Give as good an upper bound as you can for f(N).<br />
<br />
Note that, despite the evident regularities you'll observe in this problem, we do not even know whether there are infinitely many primes of the form x^2 + 1! You would become very famous if you proved this.<br />
<br />
* '''Feb 7''': 2.6 (the formulation of numerical evidence should be done by Sage if you've got Sage working, and by calculator if not; you can use an online tool like [http://primes.utm.edu/curios/includes/primetest.php this] to test whether a number is prime.) 2.8,2.9,2.11,2.12,2.14,2.19<br />
<br />
* '''Feb 17''': 2.15, 2.18, 2.23, 2.26, 2.30.<br />
<br />
Problem A: Prove that if n=pq, with p,q prime, then n is not a Carmichael number.<br />
<br />
* '''Feb 21''': Book problems: 3.4, 3.5, 3.6<br />
<br />
Problem A. Prove that there are infinitely many primes p such that 2 is '''not''' a primitive root in Z/pZ. We break this up into steps.<br />
<br />
Problem A.1. Prove that, if x is an element of Z/nZ, then x^2 is not a primitive root.<br />
<br />
Problem A.2. Prove that there are infinitely many primes p such that 2 is a square in Z/pZ. Hint: suppose there are only finitely many such primes p_1, .. p_r, and define N = (p_1 .. p_r)^2 - 2. Where can you go from here...?<br />
<br />
Problem A.3. Give a list of five primes p such that 2 is not a primitive root in Z/pZ (you can use the method of this proof or any other.)<br />
<br />
Problem B. Prove that 24 is the largest integer n such that every element of (Z/nZ)^* is a root of x^2-1.<br />
<br />
* '''Feb 28''': Problem A. Using p = 23 and q=31, show how to encrypt the message x=240 with Rabin's algorithm. Find all possible decryptions of your encrypted message.<br />
<br />
* '''March 23''': 4.1 from the book.<br />
<br />
Problem A. Give a prime factorization of the Gaussian integer 7+9i.<br />
<br />
Problem B. Read the notes from [http://www.math.ucsd.edu/~jverstra/Gaussian1.pdf here]. Note that Z[i] satisfies a ''reduction theorem'': if n and d are Gaussian integers, then there exists integers q and r such that n = qd + r and Norm(r) < Norm(d). But (by contrast with the case of Z) this d may not be unique. In some contexts it is better to be able to choose r uniquely, even if this means letting r have norm greater than Norm(d). <br />
<br />
B.1. When d = 1+2i, show that, for each n in Z[i], there is a '''unique''' pair (q,r) in Z[i] such that n = qd+r and r is contained in the set {0,1,2,3,4}. For instance, i can be written as i(1+2i) + 2, so we say i reduces to 2 mod (1+2i).<br />
(Hint: Suppose q(1+2i)+r = q'(1+2i) + r'. What can we say about (r-r'), and why is this incompatible with both r and r' being in {0,1,2,3,4}?<br />
<br />
B.2. Show that if n is an integer in Z, the reduction of n mod (1+2i) is equal to its reduction mod 5.<br />
<br />
Problem C. Let's try to figure out how to define "phi(d)" for a Gaussian integer d. Suppose S is a set of Gaussian integers such that every n in Z[i] can be written uniquely as qd+r, with q in Z[i] and r in S. (So for instance when d=1+2i, we showed in problem B that we can take S to be {0...4}. It would also be OK to take S to be {1..5} or {0,i,2i,1+i,1+2i}. In fact, it turns out that S has to be a set of size Norm(d) (I might or might not prove this in class; if not, feel free just to accept it.)<br />
<br />
Now define phi(d) to be the number of elements s of S such that s and d are coprime.<br />
<br />
C.1. Compute phi(1+2i) and phi(3).<br />
<br />
C.2. Prove that the value of phi(d) does not depend on the choice of S.<br />
<br />
C.3. Prove that every n in Z[i] which is coprime to 3 satisfies n^phi(3) = 1 mod 3; that is, Euler's theorem holds in this case. (You can prove this by direct computation; of course, if you want, you are welcome to prove that Euler's theorem holds for Z[i] in general, adapting the proof in Stein's book or the one we gave in class.)<br />
<br />
*'''March 30''' <br />
<br />
Problem A. Express 50005 as the sum of two squares.<br />
<br />
In the next two problems we denote by r(n) the number of ways to express n as the sum of two squares (i.e. the number of ordered pairs (a,b) such that a^2 + b^2 = n.) For instance, r(5) = 8: (a,b) = (+/-1, +/-2) or (+/-2, +/-1).<br />
<br />
Problem B. Prove that, for any N, there exists an integer n such that r(n) > N. (I.E., the function r(n) is ''unbounded.'')<br />
<br />
Problem C. If you like Sage, write a short program in Sage to compute r(n) and compute the average of r(n) as n ranges from 1 to 1000. Whether or not you like Sage, make a guess as to how this average would behave if you replaced "1000" by a larger and larger number. (Feel free to ask the Sage-lovers what answer they got in the optional first part of the question.) Can you prove this guess is correct?<br />
<br />
Problem D. We saw in class that the ring Z[sqrt(-5)] doesn't have unique factorization; 6 can be factored as 2*3 or (1+sqrt(-5))(1-sqrt(-5)). In this problem, we will prove that Z[sqrt(-d)] fails to have unique factorization for EVERY odd d >= 5. (Actually it's true for all d >= 5 but to make the proof manageable we'll restrict to the odd case.]<br />
<br />
D.1. Show that (1+sqrt(-d)), (1-sqrt(-d)) and 2 are irreducible in Z[sqrt(-d)].<br />
<br />
D.2. Now give an element of Z[sqrt(-d)] that has two distinct factorizations into irreducibles. (Hint: imitate the example we used for Z[sqrt(-5)].)<br />
''Remark:'' The rings Z[sqrt(d)], where d is positive, are quite different -- here we believe that there are infinitely many with unique factorization, though this conjecture has remained unproved for many decades!<br />
<br />
<!--<br />
*'''November 16'''<br />
<br />
Book problems: 5.3,5.4,5.5<br />
<br />
Problem A: We discussed in class the problem of studying which positive integers are the sum of two squares. In this problem we prove that every element n of (Z/pZ) is the sum of two squares. We argue as follows. Let S be the set of squares in (Z/pZ), and let T be the set of (Z/pZ) consisting of all elements of the form n - x^2, for some x in (Z/pZ).<br />
<br />
A.1. What is the size of S and of T? Use this to show that S and T are not disjoint.<br />
<br />
A.2. Given that S and T are not disjoint, prove that n is the sum of two squares.<br />
<br />
Problem B. Using the continued fraction expansion, find a solution to the Pell equation x^2 - 13 y^2 = 1.<br />
<br />
Problem C. Show that the modified Pell equation x^2 - 7y^2 = -1 has no solutions in integers x,y. (Hint: reduce the equation modulo a suitably chosen prime.)<br />
<br />
Problem D. Stuffy Stirnweiss finished the 1945 season with a batting average of .3085443. Using continued fractions, guess how many at-bats he had. Tony Cuccinello had a batting average of .3084577. Given that he had more than 200 and fewer than 600 at-bats, can you estimate the number of at-bats he had?<br />
<br />
'''December 7'''<br />
<br />
Problem A. When p is a prime congruent to 3 mod 4, prove that ((p-1)/2)! is either 1 or -1 in (Z/pZ). '''OPTIONAL:''' Use sage to compute whether this factorial is 1 or -1 for many primes p. Is there any pattern? Does it seem to be 1 half the time and -1 half the time? Note that I have no idea what the answer to this question is.<br />
<br />
Problem B. Using the continued fraction expansion, find a solution to the Pell equation x^2 - 13 y^2 = 1.<br />
<br />
Problem C. Show that the modified Pell equation x^2 - 7y^2 = -1 has no solutions in integers x,y. (Hint: reduce the equation modulo a suitably chosen prime.)<br />
<br />
Problem D.1. Pick two values of a in F_11 = Z/11Z (a not equal to 3), such that the equation y^2 = x^3+ax+1 defines an elliptic curve (i.e., it is smooth). For each such a, determine the number of points #E(F_11) and check that it falls inside the interval described in class. <br />
<br />
Problem D.2. The point P = (0,1) lies on each of these curves. For a =3 (the curve discussed in class) determine the order of P in the elliptic curve group, that is, find the smallest positive integer n such that nP = (infinity) -- the identity element in the group law.<br />
<br />
<!-- '''November 19'''<br />
<br />
Problem A. Using the method discussed in class (which is also the method of problem 4.6 in Stein, which in retrospect I think was too hard to assign with no preparation) find a nontrivial solution with y > 0 to the equation x^2 + 11y^2 = z^2. <br />
Problem B. Let f(X) be the number of solutions of A^2 + B^2 = C^3 such that A^2, B^2, and C^3 are all at most X. Use the heuristic described in class to explain why one might expect f(X) to grow more or less like X^{1/3}.<br />
<br />
Problem C. Show that if A+Bi is the cube of a Gaussian integer, then A^2 + B^2 is a perfect cube.<br />
<br />
Do ONE of problem D1 and D2; D1 is for people who like Sage, D2 is for people who like proving things.<br />
<br />
D1. Use Sage to compute f(X) for X = 1000, 10000, 100000. Does the answer look consistent with the heuristic prediction you made in B? Does f(X)/X^{1/3} appear to be approaching a limit?<br />
D2. We will use problem C to give a lower bound for f(X). Use this to show that there are at least X^{1/3} Gaussian integers with norm at most X that are perfect cubes. From here, show that f(X) > X^{1/3}.<br />
<br />
OPTIONAL: Show that the converse of problem C also holds -- A+Bi has norm a perfect cube if and only if A+Bi is a perfect cube. (You will need unique factorization of Gaussian integers.) Using this fact, prove that f(X) is bounded above by C X^{1/3} for some constant C.<br />
<br />
'''December 8 (note nonstandard due date)'''<br />
<br />
Problem A: Hyperbolas, ellipses, and "magic slopes"<br />
<br />
In the usual analytic geometry, both hyperbolas and ellipses are given by equations of the form<br />
<br />
ax^2 + bxy + cy^2 = d (*)<br />
<br />
<br />
(we always assume d is nonzero.)<br />
<br />
How can we tell whether such an equation describes an ellipse or a<br />
hyperbola? Well, in the geometric setting we all know and love (i.e. in<br />
R^2) a hyperbola has asymptotes and an ellipse does not. What is an <br />
asymptote? You could say it's "a line which doesn't intersect the curve", <br />
but ellipses have such lines too. You might want to say "a line which <br />
doesn't intersect the curve but comes closer and closer to the curve," but <br />
the problem here is that this a) is somewhat imprecise, and b) relies on a <br />
notion of "closer" that is not going to be very clear in Z/pZ, which is <br />
our ultimate goal!<br />
<br />
<br />
So let me state it a different way: let's say the asymptotes of a <br />
hyperbola have slopes m_1 and m_2. These are "magic slopes" in the <br />
following sense: ANY line of slope m_1 (and ditto for m_2) intersects the <br />
hyperbola in at most one point. An ellipse doesn't have "magic slopes" -- <br />
you can convince yourself by drawing pictures that, for any slope m, you <br />
can find a line of slope m striking the ellipse twice.<br />
<br />
<br />
[REMARK: For technical reasons we're going to ignore hyperbolae with a <br />
vertical asymptote, since vertical lines don't exactly have slope.]<br />
<br />
<br />
Now you might know the criterion that, over the real numbers.<br />
<br />
<br />
ax^2 + bxy + cy^2 = d<br />
<br />
<br />
is a hyperbola if b^2 - 4ac > 0 and an ellipse if b^2 - 4ac < 0. (For <br />
instance, x^2 - y^2 = 1 is a hyperbola with magic slopes 1 and -1, and x^2 <br />
+ y^2 = 1 is an ellipse.) <br />
This criterion does not make sense in Z/pZ, where there is no notion of <br />
"greater" or "less."<br />
<br />
<br />
QUESTION A.1.<br />
<br />
For which primes p is it the case that <br />
<br />
x^2 + y^2 = 1<br />
<br />
has magic slopes in Z/pZ? <br />
<br />
QUESTION A.2. For which primes p is it the case that<br />
<br />
<br />
x^2 - y^2 = 1<br />
<br />
<br />
has magic slopes in Z/pZ?<br />
<br />
<br />
QUESTION B: Back in the very first week of this course we studied the set of integers x such that x^2 + 1 is prime. Based on the heuristics we discussed in class, about how many x between 1 and N would you expect to satisfy "x^2 + 1 is prime?"<br />
<br />
<br />
QUESTION C: Let z be a complex number. Then the set of all complex numbers of the form a + bz, with a and b integers, forms a lattice in the complex plane (thought of as R^2.) What is the covolume of this lattice, in terms of z?<br />
<br />
<br />
QUESTION D: Let z be a complex number. Using Minkowski's theorem and the result of question C, show that there exist integers a,b, not both zero, such that Norm(a+bz) is at most (4/pi) Im(z). (Hint: let Omega be the set of all complex numbers of norm at most (4/pi) Im(z).)<br />
<br />
<br />
OPTIONAL EXTRA: Can you give an exact formula for the minimal norm of any nonzero complex number of the form a+bz?<br />
<br />
'''December 12''' Book problems 6.1, 6.2, 6.5, 6.10. For extra credit attempt to do problem 6.9.<br />
<br />
--></div>Andreichttps://www.math.wisc.edu/wiki/index.php?title=Math_567_--_Elementary_Number_Theory&diff=19327Math 567 -- Elementary Number Theory2020-04-02T13:55:02Z<p>Andreic: </p>
<hr />
<div>'''MATH 567<br />
<br />
Elementary Number Theory'''<br />
<br />
MWF 1:20-2:10, Van Vleck B119<br />
<br />
'''Professor:''' [http://www.math.wisc.edu/~andreic/ Andrei Caldararu] (andreic@math.wisc.edu)<br />
<br />
''Office Hours:'' Wednesdays 2:30-3:30, Van Vleck 605. <br />
<br />
'''Grader:''' Yifan Peng, email: peng64@wisc.edu<br />
<br />
''Office Hours:'' TBA.<br />
<br />
Math 567 is a course in elementary number theory, aimed at undergraduates majoring in math or other quantitative disciplines. A general familiarity with abstract algebra at the level of Math 541 will be assumed, but students who haven't taken 541 are welcome to attend if they're willing to play a little catchup. We will be using William Stein's new (and cheap) textbook [http://www.amazon.com/Elementary-Number-Theory-Computational-Undergraduate/dp/0387855246 Elementary Number Theory: Primes, Congruences, and Secrets], which emphasizes computational approaches to the subject. If you don't need a physical copy of the book, [http://www.williamstein.org/ent/ it is available as a free legal .pdf.] We will be using the (free, public-domain) mathematical software [http://www.sagemath.org/ SAGE], developed largely by Stein, as an integral component of our coursework. There is a [http://doc.sagemath.org/pdf/en/tutorial/SageTutorial.pdf useful online tutorial.] You can download SAGE to your own computer or [http://www.sagenb.org use it online].<br />
<br />
Topics include some subset of, but are not limited to: Divisibility, the Euclidean algorithm and the GCD, linear Diophantine equations, prime numbers and uniqueness of factorization. Congruences, Chinese remainder theorem, Fermat's "little" theorem, Wilson's theorem, Euler's theorem and totient function, the RSA cryptosystem, Rabin's encryption scheme, Diffie-Hellman key exchange protocol. Number-theoretic functions, multiplicative functions, Möbius inversion. Primitive roots and indices. Quadratic reciprocity and the Legendre symbol. Perfect numbers, Mersenne primes, Fermat primes. Pythagorean triples, Special cases of Fermat's "last" theorem. Fibonacci numbers. Continued fractions. Distribution of primes, discussion of prime number theorem. Primality testing and factoring algorithms.<br />
<br />
'''Course Policies:''' Homework will be due on Fridays. It can be turned in late only with ''advance'' permission from your grader. It is acceptable to use calculators and computers on homework (indeed, some of it will require a computer) but calculators are not allowed during exams. You are encouraged to work together on homework, but writeups must be done individually.<br />
<br />
Many of the problems in this course will ask you to prove things. I expect proofs to be written in English sentences; the proofs in Stein's book are a good model for the level of verbosity I am looking for. <br />
<br />
'''Grading:''' The grade in Math 567 will be composed of 50% homework, 25% midterm, 25% final. The midterm will be on March 11, in class. The final exam will be on 5/8/2020, 7:45AM-9:45AM in room TBA.<br />
<br />
'''Syllabus:''' <br />
(This may change as we see what pace works well for the course. All section numbers refer to Stein's book.)<br />
<br />
* Jan 22-31: Prime numbers, prime factorizations, Euclidean algorithm and GCD (1.1-1.2)<br />
* Feb 3-7: The integers mod n, Euler's theorem, the phi function (2.1-2.2)<br />
* Feb 10-14: Modular exponentiation, primality testing, and primitive roots (2.4-2.5)<br />
* Feb 17-21: Public-key cryptography and RSA (3.1-3.4)<br />
* Feb 24-28: Rabin's algorithm (not in the book); algebraic numbers <br />
* Mar 2-6: Quadratic reciprocity (4.1-4.4)<br />
* Mar 9, Mar 13: Finite and infinite continued fractions (5.1-5.3)<br />
* Mar 11: ''Midterm exam''<br />
* Mar 23-30: Continued fractions and diophantine approximation (5.4-5.5)<br />
* Apr 1-3: Diophantine equations I: Pell's equation and Lagrange's theorem<br />
* Apr 6-10: More diophantine equations, elliptic curves (6.1)<br />
* Apr 13-17: Applications of elliptic curves (6.2-6.3)<br />
* Apr 20-24: More applications of elliptic curves (6.4)<br />
* Apr 27-May 1: advanced topic TBD: maybe additional discussion of cryptographic techniques?<br />
<br />
'''Video Lectures:'''<br />
Due to the university being transitioned to online teaching only, out lectures after spring break will need to be delivered online. I will post here videos for you to watch for the material we need to cover. Please watch them, and please give me feedback on how they could be improved.<br />
<br />
* '''Lecture for Mar. 23''': watch it [https://mediaspace.wisc.edu/media/Math+567+March+23+Lecture/0_fkhvkxxv here].<br />
* '''Lecture for Mar. 25''': watch it [https://mediaspace.wisc.edu/media/Math+567+March+25+lecture/0_qgkhpm19 here].<br />
* '''Lecture for Mar. 27''': watch it [https://mediaspace.wisc.edu/media/March+27+Math+567+Lecture+Video/0_ajbr2ja0 here].<br />
* '''Lecture for Mar. 30''': watch it [https://mediaspace.wisc.edu/media/March+30+Math+567+Lecture/0_yz44sw2c here].<br />
* '''Lecture for Apr. 1''': watch it [https://mediaspace.wisc.edu/media/April+1+Math+567/0_e64bdfcf here].<br />
* '''Lecture for Apr. 3''': watch it [https://mediaspace.wisc.edu/media/April+3+Math+567+Video/0_02rvfino here].<br />
<br />
'''Homework:'''<br />
Homework is due at the beginning of class on the specified Friday. Typing your homework is not a requirement, but if you don't already know LaTeX I highly recommend that you learn it and use it to typeset your homework. I will sometimes assign extra problems, which I will e-mail to the class list and include here.<br />
<br />
* '''Jan 31''': 1.1, 1.3, 1.5, 1.7 (use SAGE), 1.8, 1.14.<br />
Problem A: Use SAGE to compute the number of x in [1..N] such that x^2 + 1 is prime, for N = 100, N = 1000, and N = 10000. Let f(N) be the number of such x. <br />
<br />
a) Can you formulate a conjecture about the relationship between f(N) and N/log N? <br />
<br />
b) What if x^2 + 1 is replaced with x^2 + 2? Can you explain why x^2 + 2 appears less likely to be prime? (Hint: consider x mod 3.)<br />
<br />
c) Prove that f(N) is at most (1/2)N+1. (Hint: consider x mod 2.) <br />
<br />
d) Give as good an upper bound as you can for f(N).<br />
<br />
Note that, despite the evident regularities you'll observe in this problem, we do not even know whether there are infinitely many primes of the form x^2 + 1! You would become very famous if you proved this.<br />
<br />
* '''Feb 7''': 2.6 (the formulation of numerical evidence should be done by Sage if you've got Sage working, and by calculator if not; you can use an online tool like [http://primes.utm.edu/curios/includes/primetest.php this] to test whether a number is prime.) 2.8,2.9,2.11,2.12,2.14,2.19<br />
<br />
* '''Feb 17''': 2.15, 2.18, 2.23, 2.26, 2.30.<br />
<br />
Problem A: Prove that if n=pq, with p,q prime, then n is not a Carmichael number.<br />
<br />
* '''Feb 21''': Book problems: 3.4, 3.5, 3.6<br />
<br />
Problem A. Prove that there are infinitely many primes p such that 2 is '''not''' a primitive root in Z/pZ. We break this up into steps.<br />
<br />
Problem A.1. Prove that, if x is an element of Z/nZ, then x^2 is not a primitive root.<br />
<br />
Problem A.2. Prove that there are infinitely many primes p such that 2 is a square in Z/pZ. Hint: suppose there are only finitely many such primes p_1, .. p_r, and define N = (p_1 .. p_r)^2 - 2. Where can you go from here...?<br />
<br />
Problem A.3. Give a list of five primes p such that 2 is not a primitive root in Z/pZ (you can use the method of this proof or any other.)<br />
<br />
Problem B. Prove that 24 is the largest integer n such that every element of (Z/nZ)^* is a root of x^2-1.<br />
<br />
* '''Feb 28''': Problem A. Using p = 23 and q=31, show how to encrypt the message x=240 with Rabin's algorithm. Find all possible decryptions of your encrypted message.<br />
<br />
* '''March 23''': 4.1 from the book.<br />
<br />
Problem A. Give a prime factorization of the Gaussian integer 7+9i.<br />
<br />
Problem B. Read the notes from [http://www.math.ucsd.edu/~jverstra/Gaussian1.pdf here]. Note that Z[i] satisfies a ''reduction theorem'': if n and d are Gaussian integers, then there exists integers q and r such that n = qd + r and Norm(r) < Norm(d). But (by contrast with the case of Z) this d may not be unique. In some contexts it is better to be able to choose r uniquely, even if this means letting r have norm greater than Norm(d). <br />
<br />
B.1. When d = 1+2i, show that, for each n in Z[i], there is a '''unique''' pair (q,r) in Z[i] such that n = qd+r and r is contained in the set {0,1,2,3,4}. For instance, i can be written as i(1+2i) + 2, so we say i reduces to 2 mod (1+2i).<br />
(Hint: Suppose q(1+2i)+r = q'(1+2i) + r'. What can we say about (r-r'), and why is this incompatible with both r and r' being in {0,1,2,3,4}?<br />
<br />
B.2. Show that if n is an integer in Z, the reduction of n mod (1+2i) is equal to its reduction mod 5.<br />
<br />
Problem C. Let's try to figure out how to define "phi(d)" for a Gaussian integer d. Suppose S is a set of Gaussian integers such that every n in Z[i] can be written uniquely as qd+r, with q in Z[i] and r in S. (So for instance when d=1+2i, we showed in problem B that we can take S to be {0...4}. It would also be OK to take S to be {1..5} or {0,i,2i,1+i,1+2i}. In fact, it turns out that S has to be a set of size Norm(d) (I might or might not prove this in class; if not, feel free just to accept it.)<br />
<br />
Now define phi(d) to be the number of elements s of S such that s and d are coprime.<br />
<br />
C.1. Compute phi(1+2i) and phi(3).<br />
<br />
C.2. Prove that the value of phi(d) does not depend on the choice of S.<br />
<br />
C.3. Prove that every n in Z[i] which is coprime to 3 satisfies n^phi(3) = 1 mod 3; that is, Euler's theorem holds in this case. (You can prove this by direct computation; of course, if you want, you are welcome to prove that Euler's theorem holds for Z[i] in general, adapting the proof in Stein's book or the one we gave in class.)<br />
<br />
*'''March 30''' <br />
<br />
Problem A. Express 50005 as the sum of two squares.<br />
<br />
In the next two problems we denote by r(n) the number of ways to express n as the sum of two squares (i.e. the number of ordered pairs (a,b) such that a^2 + b^2 = n.) For instance, r(5) = 8: (a,b) = (+/-1, +/-2) or (+/-2, +/-1).<br />
<br />
Problem B. Prove that, for any N, there exists an integer n such that r(n) > N. (I.E., the function r(n) is ''unbounded.'')<br />
<br />
Problem C. If you like Sage, write a short program in Sage to compute r(n) and compute the average of r(n) as n ranges from 1 to 1000. Whether or not you like Sage, make a guess as to how this average would behave if you replaced "1000" by a larger and larger number. (Feel free to ask the Sage-lovers what answer they got in the optional first part of the question.) Can you prove this guess is correct?<br />
<br />
Problem D. We saw in class that the ring Z[sqrt(-5)] doesn't have unique factorization; 6 can be factored as 2*3 or (1+sqrt(-5))(1-sqrt(-5)). In this problem, we will prove that Z[sqrt(-d)] fails to have unique factorization for EVERY odd d >= 5. (Actually it's true for all d >= 5 but to make the proof manageable we'll restrict to the odd case.]<br />
<br />
D.1. Show that (1+sqrt(-d)), (1-sqrt(-d)) and 2 are irreducible in Z[sqrt(-d)].<br />
<br />
D.2. Now give an element of Z[sqrt(-d)] that has two distinct factorizations into irreducibles. (Hint: imitate the example we used for Z[sqrt(-5)].)<br />
''Remark:'' The rings Z[sqrt(d)], where d is positive, are quite different -- here we believe that there are infinitely many with unique factorization, though this conjecture has remained unproved for many decades!<br />
<br />
<!--<br />
*'''November 16'''<br />
<br />
Book problems: 5.3,5.4,5.5<br />
<br />
Problem A: We discussed in class the problem of studying which positive integers are the sum of two squares. In this problem we prove that every element n of (Z/pZ) is the sum of two squares. We argue as follows. Let S be the set of squares in (Z/pZ), and let T be the set of (Z/pZ) consisting of all elements of the form n - x^2, for some x in (Z/pZ).<br />
<br />
A.1. What is the size of S and of T? Use this to show that S and T are not disjoint.<br />
<br />
A.2. Given that S and T are not disjoint, prove that n is the sum of two squares.<br />
<br />
Problem B. Using the continued fraction expansion, find a solution to the Pell equation x^2 - 13 y^2 = 1.<br />
<br />
Problem C. Show that the modified Pell equation x^2 - 7y^2 = -1 has no solutions in integers x,y. (Hint: reduce the equation modulo a suitably chosen prime.)<br />
<br />
Problem D. Stuffy Stirnweiss finished the 1945 season with a batting average of .3085443. Using continued fractions, guess how many at-bats he had. Tony Cuccinello had a batting average of .3084577. Given that he had more than 200 and fewer than 600 at-bats, can you estimate the number of at-bats he had?<br />
<br />
'''December 7'''<br />
<br />
Problem A. When p is a prime congruent to 3 mod 4, prove that ((p-1)/2)! is either 1 or -1 in (Z/pZ). '''OPTIONAL:''' Use sage to compute whether this factorial is 1 or -1 for many primes p. Is there any pattern? Does it seem to be 1 half the time and -1 half the time? Note that I have no idea what the answer to this question is.<br />
<br />
Problem B. Using the continued fraction expansion, find a solution to the Pell equation x^2 - 13 y^2 = 1.<br />
<br />
Problem C. Show that the modified Pell equation x^2 - 7y^2 = -1 has no solutions in integers x,y. (Hint: reduce the equation modulo a suitably chosen prime.)<br />
<br />
Problem D.1. Pick two values of a in F_11 = Z/11Z (a not equal to 3), such that the equation y^2 = x^3+ax+1 defines an elliptic curve (i.e., it is smooth). For each such a, determine the number of points #E(F_11) and check that it falls inside the interval described in class. <br />
<br />
Problem D.2. The point P = (0,1) lies on each of these curves. For a =3 (the curve discussed in class) determine the order of P in the elliptic curve group, that is, find the smallest positive integer n such that nP = (infinity) -- the identity element in the group law.<br />
<br />
<!-- '''November 19'''<br />
<br />
Problem A. Using the method discussed in class (which is also the method of problem 4.6 in Stein, which in retrospect I think was too hard to assign with no preparation) find a nontrivial solution with y > 0 to the equation x^2 + 11y^2 = z^2. <br />
Problem B. Let f(X) be the number of solutions of A^2 + B^2 = C^3 such that A^2, B^2, and C^3 are all at most X. Use the heuristic described in class to explain why one might expect f(X) to grow more or less like X^{1/3}.<br />
<br />
Problem C. Show that if A+Bi is the cube of a Gaussian integer, then A^2 + B^2 is a perfect cube.<br />
<br />
Do ONE of problem D1 and D2; D1 is for people who like Sage, D2 is for people who like proving things.<br />
<br />
D1. Use Sage to compute f(X) for X = 1000, 10000, 100000. Does the answer look consistent with the heuristic prediction you made in B? Does f(X)/X^{1/3} appear to be approaching a limit?<br />
D2. We will use problem C to give a lower bound for f(X). Use this to show that there are at least X^{1/3} Gaussian integers with norm at most X that are perfect cubes. From here, show that f(X) > X^{1/3}.<br />
<br />
OPTIONAL: Show that the converse of problem C also holds -- A+Bi has norm a perfect cube if and only if A+Bi is a perfect cube. (You will need unique factorization of Gaussian integers.) Using this fact, prove that f(X) is bounded above by C X^{1/3} for some constant C.<br />
<br />
'''December 8 (note nonstandard due date)'''<br />
<br />
Problem A: Hyperbolas, ellipses, and "magic slopes"<br />
<br />
In the usual analytic geometry, both hyperbolas and ellipses are given by equations of the form<br />
<br />
ax^2 + bxy + cy^2 = d (*)<br />
<br />
<br />
(we always assume d is nonzero.)<br />
<br />
How can we tell whether such an equation describes an ellipse or a<br />
hyperbola? Well, in the geometric setting we all know and love (i.e. in<br />
R^2) a hyperbola has asymptotes and an ellipse does not. What is an <br />
asymptote? You could say it's "a line which doesn't intersect the curve", <br />
but ellipses have such lines too. You might want to say "a line which <br />
doesn't intersect the curve but comes closer and closer to the curve," but <br />
the problem here is that this a) is somewhat imprecise, and b) relies on a <br />
notion of "closer" that is not going to be very clear in Z/pZ, which is <br />
our ultimate goal!<br />
<br />
<br />
So let me state it a different way: let's say the asymptotes of a <br />
hyperbola have slopes m_1 and m_2. These are "magic slopes" in the <br />
following sense: ANY line of slope m_1 (and ditto for m_2) intersects the <br />
hyperbola in at most one point. An ellipse doesn't have "magic slopes" -- <br />
you can convince yourself by drawing pictures that, for any slope m, you <br />
can find a line of slope m striking the ellipse twice.<br />
<br />
<br />
[REMARK: For technical reasons we're going to ignore hyperbolae with a <br />
vertical asymptote, since vertical lines don't exactly have slope.]<br />
<br />
<br />
Now you might know the criterion that, over the real numbers.<br />
<br />
<br />
ax^2 + bxy + cy^2 = d<br />
<br />
<br />
is a hyperbola if b^2 - 4ac > 0 and an ellipse if b^2 - 4ac < 0. (For <br />
instance, x^2 - y^2 = 1 is a hyperbola with magic slopes 1 and -1, and x^2 <br />
+ y^2 = 1 is an ellipse.) <br />
This criterion does not make sense in Z/pZ, where there is no notion of <br />
"greater" or "less."<br />
<br />
<br />
QUESTION A.1.<br />
<br />
For which primes p is it the case that <br />
<br />
x^2 + y^2 = 1<br />
<br />
has magic slopes in Z/pZ? <br />
<br />
QUESTION A.2. For which primes p is it the case that<br />
<br />
<br />
x^2 - y^2 = 1<br />
<br />
<br />
has magic slopes in Z/pZ?<br />
<br />
<br />
QUESTION B: Back in the very first week of this course we studied the set of integers x such that x^2 + 1 is prime. Based on the heuristics we discussed in class, about how many x between 1 and N would you expect to satisfy "x^2 + 1 is prime?"<br />
<br />
<br />
QUESTION C: Let z be a complex number. Then the set of all complex numbers of the form a + bz, with a and b integers, forms a lattice in the complex plane (thought of as R^2.) What is the covolume of this lattice, in terms of z?<br />
<br />
<br />
QUESTION D: Let z be a complex number. Using Minkowski's theorem and the result of question C, show that there exist integers a,b, not both zero, such that Norm(a+bz) is at most (4/pi) Im(z). (Hint: let Omega be the set of all complex numbers of norm at most (4/pi) Im(z).)<br />
<br />
<br />
OPTIONAL EXTRA: Can you give an exact formula for the minimal norm of any nonzero complex number of the form a+bz?<br />
<br />
'''December 12''' Book problems 6.1, 6.2, 6.5, 6.10. For extra credit attempt to do problem 6.9.<br />
<br />
--></div>Andreichttps://www.math.wisc.edu/wiki/index.php?title=Math_567_--_Elementary_Number_Theory&diff=19321Math 567 -- Elementary Number Theory2020-04-01T01:19:03Z<p>Andreic: </p>
<hr />
<div>'''MATH 567<br />
<br />
Elementary Number Theory'''<br />
<br />
MWF 1:20-2:10, Van Vleck B119<br />
<br />
'''Professor:''' [http://www.math.wisc.edu/~andreic/ Andrei Caldararu] (andreic@math.wisc.edu)<br />
<br />
''Office Hours:'' Wednesdays 2:30-3:30, Van Vleck 605. <br />
<br />
'''Grader:''' Yifan Peng, email: peng64@wisc.edu<br />
<br />
''Office Hours:'' TBA.<br />
<br />
Math 567 is a course in elementary number theory, aimed at undergraduates majoring in math or other quantitative disciplines. A general familiarity with abstract algebra at the level of Math 541 will be assumed, but students who haven't taken 541 are welcome to attend if they're willing to play a little catchup. We will be using William Stein's new (and cheap) textbook [http://www.amazon.com/Elementary-Number-Theory-Computational-Undergraduate/dp/0387855246 Elementary Number Theory: Primes, Congruences, and Secrets], which emphasizes computational approaches to the subject. If you don't need a physical copy of the book, [http://www.williamstein.org/ent/ it is available as a free legal .pdf.] We will be using the (free, public-domain) mathematical software [http://www.sagemath.org/ SAGE], developed largely by Stein, as an integral component of our coursework. There is a [http://doc.sagemath.org/pdf/en/tutorial/SageTutorial.pdf useful online tutorial.] You can download SAGE to your own computer or [http://www.sagenb.org use it online].<br />
<br />
Topics include some subset of, but are not limited to: Divisibility, the Euclidean algorithm and the GCD, linear Diophantine equations, prime numbers and uniqueness of factorization. Congruences, Chinese remainder theorem, Fermat's "little" theorem, Wilson's theorem, Euler's theorem and totient function, the RSA cryptosystem, Rabin's encryption scheme, Diffie-Hellman key exchange protocol. Number-theoretic functions, multiplicative functions, Möbius inversion. Primitive roots and indices. Quadratic reciprocity and the Legendre symbol. Perfect numbers, Mersenne primes, Fermat primes. Pythagorean triples, Special cases of Fermat's "last" theorem. Fibonacci numbers. Continued fractions. Distribution of primes, discussion of prime number theorem. Primality testing and factoring algorithms.<br />
<br />
'''Course Policies:''' Homework will be due on Fridays. It can be turned in late only with ''advance'' permission from your grader. It is acceptable to use calculators and computers on homework (indeed, some of it will require a computer) but calculators are not allowed during exams. You are encouraged to work together on homework, but writeups must be done individually.<br />
<br />
Many of the problems in this course will ask you to prove things. I expect proofs to be written in English sentences; the proofs in Stein's book are a good model for the level of verbosity I am looking for. <br />
<br />
'''Grading:''' The grade in Math 567 will be composed of 50% homework, 25% midterm, 25% final. The midterm will be on March 11, in class. The final exam will be on 5/8/2020, 7:45AM-9:45AM in room TBA.<br />
<br />
'''Syllabus:''' <br />
(This may change as we see what pace works well for the course. All section numbers refer to Stein's book.)<br />
<br />
* Jan 22-31: Prime numbers, prime factorizations, Euclidean algorithm and GCD (1.1-1.2)<br />
* Feb 3-7: The integers mod n, Euler's theorem, the phi function (2.1-2.2)<br />
* Feb 10-14: Modular exponentiation, primality testing, and primitive roots (2.4-2.5)<br />
* Feb 17-21: Public-key cryptography and RSA (3.1-3.4)<br />
* Feb 24-28: Rabin's algorithm (not in the book); algebraic numbers <br />
* Mar 2-6: Quadratic reciprocity (4.1-4.4)<br />
* Mar 9, Mar 13: Finite and infinite continued fractions (5.1-5.3)<br />
* Mar 11: ''Midterm exam''<br />
* Mar 23-30: Continued fractions and diophantine approximation (5.4-5.5)<br />
* Apr 1-3: Diophantine equations I: Pell's equation and Lagrange's theorem<br />
* Apr 6-10: More diophantine equations, elliptic curves (6.1)<br />
* Apr 13-17: Applications of elliptic curves (6.2-6.3)<br />
* Apr 20-24: More applications of elliptic curves (6.4)<br />
* Apr 27-May 1: advanced topic TBD: maybe additional discussion of cryptographic techniques?<br />
<br />
'''Video Lectures:'''<br />
Due to the university being transitioned to online teaching only, out lectures after spring break will need to be delivered online. I will post here videos for you to watch for the material we need to cover. Please watch them, and please give me feedback on how they could be improved.<br />
<br />
* '''Lecture for Mar. 23''': watch it [https://mediaspace.wisc.edu/media/Math+567+March+23+Lecture/0_fkhvkxxv here].<br />
* '''Lecture for Mar. 25''': watch it [https://mediaspace.wisc.edu/media/Math+567+March+25+lecture/0_qgkhpm19 here].<br />
* '''Lecture for Mar. 27''': watch it [https://mediaspace.wisc.edu/media/March+27+Math+567+Lecture+Video/0_ajbr2ja0 here].<br />
* '''Lecture for Mar. 30''': watch it [https://mediaspace.wisc.edu/media/March+30+Math+567+Lecture/0_yz44sw2c here].<br />
* '''Lecture for Apr. 1''': watch it [https://mediaspace.wisc.edu/media/April+1+Math+567/0_e64bdfcf here].<br />
<br />
'''Homework:'''<br />
Homework is due at the beginning of class on the specified Friday. Typing your homework is not a requirement, but if you don't already know LaTeX I highly recommend that you learn it and use it to typeset your homework. I will sometimes assign extra problems, which I will e-mail to the class list and include here.<br />
<br />
* '''Jan 31''': 1.1, 1.3, 1.5, 1.7 (use SAGE), 1.8, 1.14.<br />
Problem A: Use SAGE to compute the number of x in [1..N] such that x^2 + 1 is prime, for N = 100, N = 1000, and N = 10000. Let f(N) be the number of such x. <br />
<br />
a) Can you formulate a conjecture about the relationship between f(N) and N/log N? <br />
<br />
b) What if x^2 + 1 is replaced with x^2 + 2? Can you explain why x^2 + 2 appears less likely to be prime? (Hint: consider x mod 3.)<br />
<br />
c) Prove that f(N) is at most (1/2)N+1. (Hint: consider x mod 2.) <br />
<br />
d) Give as good an upper bound as you can for f(N).<br />
<br />
Note that, despite the evident regularities you'll observe in this problem, we do not even know whether there are infinitely many primes of the form x^2 + 1! You would become very famous if you proved this.<br />
<br />
* '''Feb 7''': 2.6 (the formulation of numerical evidence should be done by Sage if you've got Sage working, and by calculator if not; you can use an online tool like [http://primes.utm.edu/curios/includes/primetest.php this] to test whether a number is prime.) 2.8,2.9,2.11,2.12,2.14,2.19<br />
<br />
* '''Feb 17''': 2.15, 2.18, 2.23, 2.26, 2.30.<br />
<br />
Problem A: Prove that if n=pq, with p,q prime, then n is not a Carmichael number.<br />
<br />
* '''Feb 21''': Book problems: 3.4, 3.5, 3.6<br />
<br />
Problem A. Prove that there are infinitely many primes p such that 2 is '''not''' a primitive root in Z/pZ. We break this up into steps.<br />
<br />
Problem A.1. Prove that, if x is an element of Z/nZ, then x^2 is not a primitive root.<br />
<br />
Problem A.2. Prove that there are infinitely many primes p such that 2 is a square in Z/pZ. Hint: suppose there are only finitely many such primes p_1, .. p_r, and define N = (p_1 .. p_r)^2 - 2. Where can you go from here...?<br />
<br />
Problem A.3. Give a list of five primes p such that 2 is not a primitive root in Z/pZ (you can use the method of this proof or any other.)<br />
<br />
Problem B. Prove that 24 is the largest integer n such that every element of (Z/nZ)^* is a root of x^2-1.<br />
<br />
* '''Feb 28''': Problem A. Using p = 23 and q=31, show how to encrypt the message x=240 with Rabin's algorithm. Find all possible decryptions of your encrypted message.<br />
<br />
* '''March 23''': 4.1 from the book.<br />
<br />
Problem A. Give a prime factorization of the Gaussian integer 7+9i.<br />
<br />
Problem B. Read the notes from [http://www.math.ucsd.edu/~jverstra/Gaussian1.pdf here]. Note that Z[i] satisfies a ''reduction theorem'': if n and d are Gaussian integers, then there exists integers q and r such that n = qd + r and Norm(r) < Norm(d). But (by contrast with the case of Z) this d may not be unique. In some contexts it is better to be able to choose r uniquely, even if this means letting r have norm greater than Norm(d). <br />
<br />
B.1. When d = 1+2i, show that, for each n in Z[i], there is a '''unique''' pair (q,r) in Z[i] such that n = qd+r and r is contained in the set {0,1,2,3,4}. For instance, i can be written as i(1+2i) + 2, so we say i reduces to 2 mod (1+2i).<br />
(Hint: Suppose q(1+2i)+r = q'(1+2i) + r'. What can we say about (r-r'), and why is this incompatible with both r and r' being in {0,1,2,3,4}?<br />
<br />
B.2. Show that if n is an integer in Z, the reduction of n mod (1+2i) is equal to its reduction mod 5.<br />
<br />
Problem C. Let's try to figure out how to define "phi(d)" for a Gaussian integer d. Suppose S is a set of Gaussian integers such that every n in Z[i] can be written uniquely as qd+r, with q in Z[i] and r in S. (So for instance when d=1+2i, we showed in problem B that we can take S to be {0...4}. It would also be OK to take S to be {1..5} or {0,i,2i,1+i,1+2i}. In fact, it turns out that S has to be a set of size Norm(d) (I might or might not prove this in class; if not, feel free just to accept it.)<br />
<br />
Now define phi(d) to be the number of elements s of S such that s and d are coprime.<br />
<br />
C.1. Compute phi(1+2i) and phi(3).<br />
<br />
C.2. Prove that the value of phi(d) does not depend on the choice of S.<br />
<br />
C.3. Prove that every n in Z[i] which is coprime to 3 satisfies n^phi(3) = 1 mod 3; that is, Euler's theorem holds in this case. (You can prove this by direct computation; of course, if you want, you are welcome to prove that Euler's theorem holds for Z[i] in general, adapting the proof in Stein's book or the one we gave in class.)<br />
<br />
*'''March 30''' <br />
<br />
Problem A. Express 50005 as the sum of two squares.<br />
<br />
In the next two problems we denote by r(n) the number of ways to express n as the sum of two squares (i.e. the number of ordered pairs (a,b) such that a^2 + b^2 = n.) For instance, r(5) = 8: (a,b) = (+/-1, +/-2) or (+/-2, +/-1).<br />
<br />
Problem B. Prove that, for any N, there exists an integer n such that r(n) > N. (I.E., the function r(n) is ''unbounded.'')<br />
<br />
Problem C. If you like Sage, write a short program in Sage to compute r(n) and compute the average of r(n) as n ranges from 1 to 1000. Whether or not you like Sage, make a guess as to how this average would behave if you replaced "1000" by a larger and larger number. (Feel free to ask the Sage-lovers what answer they got in the optional first part of the question.) Can you prove this guess is correct?<br />
<br />
Problem D. We saw in class that the ring Z[sqrt(-5)] doesn't have unique factorization; 6 can be factored as 2*3 or (1+sqrt(-5))(1-sqrt(-5)). In this problem, we will prove that Z[sqrt(-d)] fails to have unique factorization for EVERY odd d >= 5. (Actually it's true for all d >= 5 but to make the proof manageable we'll restrict to the odd case.]<br />
<br />
D.1. Show that (1+sqrt(-d)), (1-sqrt(-d)) and 2 are irreducible in Z[sqrt(-d)].<br />
<br />
D.2. Now give an element of Z[sqrt(-d)] that has two distinct factorizations into irreducibles. (Hint: imitate the example we used for Z[sqrt(-5)].)<br />
''Remark:'' The rings Z[sqrt(d)], where d is positive, are quite different -- here we believe that there are infinitely many with unique factorization, though this conjecture has remained unproved for many decades!<br />
<br />
<!--<br />
*'''November 16'''<br />
<br />
Book problems: 5.3,5.4,5.5<br />
<br />
Problem A: We discussed in class the problem of studying which positive integers are the sum of two squares. In this problem we prove that every element n of (Z/pZ) is the sum of two squares. We argue as follows. Let S be the set of squares in (Z/pZ), and let T be the set of (Z/pZ) consisting of all elements of the form n - x^2, for some x in (Z/pZ).<br />
<br />
A.1. What is the size of S and of T? Use this to show that S and T are not disjoint.<br />
<br />
A.2. Given that S and T are not disjoint, prove that n is the sum of two squares.<br />
<br />
Problem B. Using the continued fraction expansion, find a solution to the Pell equation x^2 - 13 y^2 = 1.<br />
<br />
Problem C. Show that the modified Pell equation x^2 - 7y^2 = -1 has no solutions in integers x,y. (Hint: reduce the equation modulo a suitably chosen prime.)<br />
<br />
Problem D. Stuffy Stirnweiss finished the 1945 season with a batting average of .3085443. Using continued fractions, guess how many at-bats he had. Tony Cuccinello had a batting average of .3084577. Given that he had more than 200 and fewer than 600 at-bats, can you estimate the number of at-bats he had?<br />
<br />
'''December 7'''<br />
<br />
Problem A. When p is a prime congruent to 3 mod 4, prove that ((p-1)/2)! is either 1 or -1 in (Z/pZ). '''OPTIONAL:''' Use sage to compute whether this factorial is 1 or -1 for many primes p. Is there any pattern? Does it seem to be 1 half the time and -1 half the time? Note that I have no idea what the answer to this question is.<br />
<br />
Problem B. Using the continued fraction expansion, find a solution to the Pell equation x^2 - 13 y^2 = 1.<br />
<br />
Problem C. Show that the modified Pell equation x^2 - 7y^2 = -1 has no solutions in integers x,y. (Hint: reduce the equation modulo a suitably chosen prime.)<br />
<br />
Problem D.1. Pick two values of a in F_11 = Z/11Z (a not equal to 3), such that the equation y^2 = x^3+ax+1 defines an elliptic curve (i.e., it is smooth). For each such a, determine the number of points #E(F_11) and check that it falls inside the interval described in class. <br />
<br />
Problem D.2. The point P = (0,1) lies on each of these curves. For a =3 (the curve discussed in class) determine the order of P in the elliptic curve group, that is, find the smallest positive integer n such that nP = (infinity) -- the identity element in the group law.<br />
<br />
<!-- '''November 19'''<br />
<br />
Problem A. Using the method discussed in class (which is also the method of problem 4.6 in Stein, which in retrospect I think was too hard to assign with no preparation) find a nontrivial solution with y > 0 to the equation x^2 + 11y^2 = z^2. <br />
Problem B. Let f(X) be the number of solutions of A^2 + B^2 = C^3 such that A^2, B^2, and C^3 are all at most X. Use the heuristic described in class to explain why one might expect f(X) to grow more or less like X^{1/3}.<br />
<br />
Problem C. Show that if A+Bi is the cube of a Gaussian integer, then A^2 + B^2 is a perfect cube.<br />
<br />
Do ONE of problem D1 and D2; D1 is for people who like Sage, D2 is for people who like proving things.<br />
<br />
D1. Use Sage to compute f(X) for X = 1000, 10000, 100000. Does the answer look consistent with the heuristic prediction you made in B? Does f(X)/X^{1/3} appear to be approaching a limit?<br />
D2. We will use problem C to give a lower bound for f(X). Use this to show that there are at least X^{1/3} Gaussian integers with norm at most X that are perfect cubes. From here, show that f(X) > X^{1/3}.<br />
<br />
OPTIONAL: Show that the converse of problem C also holds -- A+Bi has norm a perfect cube if and only if A+Bi is a perfect cube. (You will need unique factorization of Gaussian integers.) Using this fact, prove that f(X) is bounded above by C X^{1/3} for some constant C.<br />
<br />
'''December 8 (note nonstandard due date)'''<br />
<br />
Problem A: Hyperbolas, ellipses, and "magic slopes"<br />
<br />
In the usual analytic geometry, both hyperbolas and ellipses are given by equations of the form<br />
<br />
ax^2 + bxy + cy^2 = d (*)<br />
<br />
<br />
(we always assume d is nonzero.)<br />
<br />
How can we tell whether such an equation describes an ellipse or a<br />
hyperbola? Well, in the geometric setting we all know and love (i.e. in<br />
R^2) a hyperbola has asymptotes and an ellipse does not. What is an <br />
asymptote? You could say it's "a line which doesn't intersect the curve", <br />
but ellipses have such lines too. You might want to say "a line which <br />
doesn't intersect the curve but comes closer and closer to the curve," but <br />
the problem here is that this a) is somewhat imprecise, and b) relies on a <br />
notion of "closer" that is not going to be very clear in Z/pZ, which is <br />
our ultimate goal!<br />
<br />
<br />
So let me state it a different way: let's say the asymptotes of a <br />
hyperbola have slopes m_1 and m_2. These are "magic slopes" in the <br />
following sense: ANY line of slope m_1 (and ditto for m_2) intersects the <br />
hyperbola in at most one point. An ellipse doesn't have "magic slopes" -- <br />
you can convince yourself by drawing pictures that, for any slope m, you <br />
can find a line of slope m striking the ellipse twice.<br />
<br />
<br />
[REMARK: For technical reasons we're going to ignore hyperbolae with a <br />
vertical asymptote, since vertical lines don't exactly have slope.]<br />
<br />
<br />
Now you might know the criterion that, over the real numbers.<br />
<br />
<br />
ax^2 + bxy + cy^2 = d<br />
<br />
<br />
is a hyperbola if b^2 - 4ac > 0 and an ellipse if b^2 - 4ac < 0. (For <br />
instance, x^2 - y^2 = 1 is a hyperbola with magic slopes 1 and -1, and x^2 <br />
+ y^2 = 1 is an ellipse.) <br />
This criterion does not make sense in Z/pZ, where there is no notion of <br />
"greater" or "less."<br />
<br />
<br />
QUESTION A.1.<br />
<br />
For which primes p is it the case that <br />
<br />
x^2 + y^2 = 1<br />
<br />
has magic slopes in Z/pZ? <br />
<br />
QUESTION A.2. For which primes p is it the case that<br />
<br />
<br />
x^2 - y^2 = 1<br />
<br />
<br />
has magic slopes in Z/pZ?<br />
<br />
<br />
QUESTION B: Back in the very first week of this course we studied the set of integers x such that x^2 + 1 is prime. Based on the heuristics we discussed in class, about how many x between 1 and N would you expect to satisfy "x^2 + 1 is prime?"<br />
<br />
<br />
QUESTION C: Let z be a complex number. Then the set of all complex numbers of the form a + bz, with a and b integers, forms a lattice in the complex plane (thought of as R^2.) What is the covolume of this lattice, in terms of z?<br />
<br />
<br />
QUESTION D: Let z be a complex number. Using Minkowski's theorem and the result of question C, show that there exist integers a,b, not both zero, such that Norm(a+bz) is at most (4/pi) Im(z). (Hint: let Omega be the set of all complex numbers of norm at most (4/pi) Im(z).)<br />
<br />
<br />
OPTIONAL EXTRA: Can you give an exact formula for the minimal norm of any nonzero complex number of the form a+bz?<br />
<br />
'''December 12''' Book problems 6.1, 6.2, 6.5, 6.10. For extra credit attempt to do problem 6.9.<br />
<br />
--></div>Andreichttps://www.math.wisc.edu/wiki/index.php?title=Math_567_--_Elementary_Number_Theory&diff=19315Math 567 -- Elementary Number Theory2020-03-29T23:29:22Z<p>Andreic: </p>
<hr />
<div>'''MATH 567<br />
<br />
Elementary Number Theory'''<br />
<br />
MWF 1:20-2:10, Van Vleck B119<br />
<br />
'''Professor:''' [http://www.math.wisc.edu/~andreic/ Andrei Caldararu] (andreic@math.wisc.edu)<br />
<br />
''Office Hours:'' Wednesdays 2:30-3:30, Van Vleck 605. <br />
<br />
'''Grader:''' Yifan Peng, email: peng64@wisc.edu<br />
<br />
''Office Hours:'' TBA.<br />
<br />
Math 567 is a course in elementary number theory, aimed at undergraduates majoring in math or other quantitative disciplines. A general familiarity with abstract algebra at the level of Math 541 will be assumed, but students who haven't taken 541 are welcome to attend if they're willing to play a little catchup. We will be using William Stein's new (and cheap) textbook [http://www.amazon.com/Elementary-Number-Theory-Computational-Undergraduate/dp/0387855246 Elementary Number Theory: Primes, Congruences, and Secrets], which emphasizes computational approaches to the subject. If you don't need a physical copy of the book, [http://www.williamstein.org/ent/ it is available as a free legal .pdf.] We will be using the (free, public-domain) mathematical software [http://www.sagemath.org/ SAGE], developed largely by Stein, as an integral component of our coursework. There is a [http://doc.sagemath.org/pdf/en/tutorial/SageTutorial.pdf useful online tutorial.] You can download SAGE to your own computer or [http://www.sagenb.org use it online].<br />
<br />
Topics include some subset of, but are not limited to: Divisibility, the Euclidean algorithm and the GCD, linear Diophantine equations, prime numbers and uniqueness of factorization. Congruences, Chinese remainder theorem, Fermat's "little" theorem, Wilson's theorem, Euler's theorem and totient function, the RSA cryptosystem, Rabin's encryption scheme, Diffie-Hellman key exchange protocol. Number-theoretic functions, multiplicative functions, Möbius inversion. Primitive roots and indices. Quadratic reciprocity and the Legendre symbol. Perfect numbers, Mersenne primes, Fermat primes. Pythagorean triples, Special cases of Fermat's "last" theorem. Fibonacci numbers. Continued fractions. Distribution of primes, discussion of prime number theorem. Primality testing and factoring algorithms.<br />
<br />
'''Course Policies:''' Homework will be due on Fridays. It can be turned in late only with ''advance'' permission from your grader. It is acceptable to use calculators and computers on homework (indeed, some of it will require a computer) but calculators are not allowed during exams. You are encouraged to work together on homework, but writeups must be done individually.<br />
<br />
Many of the problems in this course will ask you to prove things. I expect proofs to be written in English sentences; the proofs in Stein's book are a good model for the level of verbosity I am looking for. <br />
<br />
'''Grading:''' The grade in Math 567 will be composed of 50% homework, 25% midterm, 25% final. The midterm will be on March 11, in class. The final exam will be on 5/8/2020, 7:45AM-9:45AM in room TBA.<br />
<br />
'''Syllabus:''' <br />
(This may change as we see what pace works well for the course. All section numbers refer to Stein's book.)<br />
<br />
* Jan 22-31: Prime numbers, prime factorizations, Euclidean algorithm and GCD (1.1-1.2)<br />
* Feb 3-7: The integers mod n, Euler's theorem, the phi function (2.1-2.2)<br />
* Feb 10-14: Modular exponentiation, primality testing, and primitive roots (2.4-2.5)<br />
* Feb 17-21: Public-key cryptography and RSA (3.1-3.4)<br />
* Feb 24-28: Rabin's algorithm (not in the book); algebraic numbers <br />
* Mar 2-6: Quadratic reciprocity (4.1-4.4)<br />
* Mar 9, Mar 13: Finite and infinite continued fractions (5.1-5.3)<br />
* Mar 11: ''Midterm exam''<br />
* Mar 23-30: Continued fractions and diophantine approximation (5.4-5.5)<br />
* Apr 1-3: Diophantine equations I: Pell's equation and Lagrange's theorem<br />
* Apr 6-10: More diophantine equations, elliptic curves (6.1)<br />
* Apr 13-17: Applications of elliptic curves (6.2-6.3)<br />
* Apr 20-24: More applications of elliptic curves (6.4)<br />
* Apr 27-May 1: advanced topic TBD: maybe additional discussion of cryptographic techniques?<br />
<br />
'''Video Lectures:'''<br />
Due to the university being transitioned to online teaching only, out lectures after spring break will need to be delivered online. I will post here videos for you to watch for the material we need to cover. Please watch them, and please give me feedback on how they could be improved.<br />
<br />
* '''Lecture for Mar. 23''': watch it [https://mediaspace.wisc.edu/media/Math+567+March+23+Lecture/0_fkhvkxxv here].<br />
* '''Lecture for Mar. 25''': watch it [https://mediaspace.wisc.edu/media/Math+567+March+25+lecture/0_qgkhpm19 here].<br />
* '''Lecture for Mar. 27''': watch it [https://mediaspace.wisc.edu/media/March+27+Math+567+Lecture+Video/0_ajbr2ja0 here].<br />
* '''Lecture for Mar. 30''': watch it [https://mediaspace.wisc.edu/media/March+30+Math+567+Lecture/0_yz44sw2c here].<br />
<br />
'''Homework:'''<br />
Homework is due at the beginning of class on the specified Friday. Typing your homework is not a requirement, but if you don't already know LaTeX I highly recommend that you learn it and use it to typeset your homework. I will sometimes assign extra problems, which I will e-mail to the class list and include here.<br />
<br />
* '''Jan 31''': 1.1, 1.3, 1.5, 1.7 (use SAGE), 1.8, 1.14.<br />
Problem A: Use SAGE to compute the number of x in [1..N] such that x^2 + 1 is prime, for N = 100, N = 1000, and N = 10000. Let f(N) be the number of such x. <br />
<br />
a) Can you formulate a conjecture about the relationship between f(N) and N/log N? <br />
<br />
b) What if x^2 + 1 is replaced with x^2 + 2? Can you explain why x^2 + 2 appears less likely to be prime? (Hint: consider x mod 3.)<br />
<br />
c) Prove that f(N) is at most (1/2)N+1. (Hint: consider x mod 2.) <br />
<br />
d) Give as good an upper bound as you can for f(N).<br />
<br />
Note that, despite the evident regularities you'll observe in this problem, we do not even know whether there are infinitely many primes of the form x^2 + 1! You would become very famous if you proved this.<br />
<br />
* '''Feb 7''': 2.6 (the formulation of numerical evidence should be done by Sage if you've got Sage working, and by calculator if not; you can use an online tool like [http://primes.utm.edu/curios/includes/primetest.php this] to test whether a number is prime.) 2.8,2.9,2.11,2.12,2.14,2.19<br />
<br />
* '''Feb 17''': 2.15, 2.18, 2.23, 2.26, 2.30.<br />
<br />
Problem A: Prove that if n=pq, with p,q prime, then n is not a Carmichael number.<br />
<br />
* '''Feb 21''': Book problems: 3.4, 3.5, 3.6<br />
<br />
Problem A. Prove that there are infinitely many primes p such that 2 is '''not''' a primitive root in Z/pZ. We break this up into steps.<br />
<br />
Problem A.1. Prove that, if x is an element of Z/nZ, then x^2 is not a primitive root.<br />
<br />
Problem A.2. Prove that there are infinitely many primes p such that 2 is a square in Z/pZ. Hint: suppose there are only finitely many such primes p_1, .. p_r, and define N = (p_1 .. p_r)^2 - 2. Where can you go from here...?<br />
<br />
Problem A.3. Give a list of five primes p such that 2 is not a primitive root in Z/pZ (you can use the method of this proof or any other.)<br />
<br />
Problem B. Prove that 24 is the largest integer n such that every element of (Z/nZ)^* is a root of x^2-1.<br />
<br />
* '''Feb 28''': Problem A. Using p = 23 and q=31, show how to encrypt the message x=240 with Rabin's algorithm. Find all possible decryptions of your encrypted message.<br />
<br />
* '''March 23''': 4.1 from the book.<br />
<br />
Problem A. Give a prime factorization of the Gaussian integer 7+9i.<br />
<br />
Problem B. Read the notes from [http://www.math.ucsd.edu/~jverstra/Gaussian1.pdf here]. Note that Z[i] satisfies a ''reduction theorem'': if n and d are Gaussian integers, then there exists integers q and r such that n = qd + r and Norm(r) < Norm(d). But (by contrast with the case of Z) this d may not be unique. In some contexts it is better to be able to choose r uniquely, even if this means letting r have norm greater than Norm(d). <br />
<br />
B.1. When d = 1+2i, show that, for each n in Z[i], there is a '''unique''' pair (q,r) in Z[i] such that n = qd+r and r is contained in the set {0,1,2,3,4}. For instance, i can be written as i(1+2i) + 2, so we say i reduces to 2 mod (1+2i).<br />
(Hint: Suppose q(1+2i)+r = q'(1+2i) + r'. What can we say about (r-r'), and why is this incompatible with both r and r' being in {0,1,2,3,4}?<br />
<br />
B.2. Show that if n is an integer in Z, the reduction of n mod (1+2i) is equal to its reduction mod 5.<br />
<br />
Problem C. Let's try to figure out how to define "phi(d)" for a Gaussian integer d. Suppose S is a set of Gaussian integers such that every n in Z[i] can be written uniquely as qd+r, with q in Z[i] and r in S. (So for instance when d=1+2i, we showed in problem B that we can take S to be {0...4}. It would also be OK to take S to be {1..5} or {0,i,2i,1+i,1+2i}. In fact, it turns out that S has to be a set of size Norm(d) (I might or might not prove this in class; if not, feel free just to accept it.)<br />
<br />
Now define phi(d) to be the number of elements s of S such that s and d are coprime.<br />
<br />
C.1. Compute phi(1+2i) and phi(3).<br />
<br />
C.2. Prove that the value of phi(d) does not depend on the choice of S.<br />
<br />
C.3. Prove that every n in Z[i] which is coprime to 3 satisfies n^phi(3) = 1 mod 3; that is, Euler's theorem holds in this case. (You can prove this by direct computation; of course, if you want, you are welcome to prove that Euler's theorem holds for Z[i] in general, adapting the proof in Stein's book or the one we gave in class.)<br />
<br />
*'''March 30''' <br />
<br />
Problem A. Express 50005 as the sum of two squares.<br />
<br />
In the next two problems we denote by r(n) the number of ways to express n as the sum of two squares (i.e. the number of ordered pairs (a,b) such that a^2 + b^2 = n.) For instance, r(5) = 8: (a,b) = (+/-1, +/-2) or (+/-2, +/-1).<br />
<br />
Problem B. Prove that, for any N, there exists an integer n such that r(n) > N. (I.E., the function r(n) is ''unbounded.'')<br />
<br />
Problem C. If you like Sage, write a short program in Sage to compute r(n) and compute the average of r(n) as n ranges from 1 to 1000. Whether or not you like Sage, make a guess as to how this average would behave if you replaced "1000" by a larger and larger number. (Feel free to ask the Sage-lovers what answer they got in the optional first part of the question.) Can you prove this guess is correct?<br />
<br />
Problem D. We saw in class that the ring Z[sqrt(-5)] doesn't have unique factorization; 6 can be factored as 2*3 or (1+sqrt(-5))(1-sqrt(-5)). In this problem, we will prove that Z[sqrt(-d)] fails to have unique factorization for EVERY odd d >= 5. (Actually it's true for all d >= 5 but to make the proof manageable we'll restrict to the odd case.]<br />
<br />
D.1. Show that (1+sqrt(-d)), (1-sqrt(-d)) and 2 are irreducible in Z[sqrt(-d)].<br />
<br />
D.2. Now give an element of Z[sqrt(-d)] that has two distinct factorizations into irreducibles. (Hint: imitate the example we used for Z[sqrt(-5)].)<br />
''Remark:'' The rings Z[sqrt(d)], where d is positive, are quite different -- here we believe that there are infinitely many with unique factorization, though this conjecture has remained unproved for many decades!<br />
<br />
<!--<br />
*'''November 16'''<br />
<br />
Book problems: 5.3,5.4,5.5<br />
<br />
Problem A: We discussed in class the problem of studying which positive integers are the sum of two squares. In this problem we prove that every element n of (Z/pZ) is the sum of two squares. We argue as follows. Let S be the set of squares in (Z/pZ), and let T be the set of (Z/pZ) consisting of all elements of the form n - x^2, for some x in (Z/pZ).<br />
<br />
A.1. What is the size of S and of T? Use this to show that S and T are not disjoint.<br />
<br />
A.2. Given that S and T are not disjoint, prove that n is the sum of two squares.<br />
<br />
Problem B. Using the continued fraction expansion, find a solution to the Pell equation x^2 - 13 y^2 = 1.<br />
<br />
Problem C. Show that the modified Pell equation x^2 - 7y^2 = -1 has no solutions in integers x,y. (Hint: reduce the equation modulo a suitably chosen prime.)<br />
<br />
Problem D. Stuffy Stirnweiss finished the 1945 season with a batting average of .3085443. Using continued fractions, guess how many at-bats he had. Tony Cuccinello had a batting average of .3084577. Given that he had more than 200 and fewer than 600 at-bats, can you estimate the number of at-bats he had?<br />
<br />
'''December 7'''<br />
<br />
Problem A. When p is a prime congruent to 3 mod 4, prove that ((p-1)/2)! is either 1 or -1 in (Z/pZ). '''OPTIONAL:''' Use sage to compute whether this factorial is 1 or -1 for many primes p. Is there any pattern? Does it seem to be 1 half the time and -1 half the time? Note that I have no idea what the answer to this question is.<br />
<br />
Problem B. Using the continued fraction expansion, find a solution to the Pell equation x^2 - 13 y^2 = 1.<br />
<br />
Problem C. Show that the modified Pell equation x^2 - 7y^2 = -1 has no solutions in integers x,y. (Hint: reduce the equation modulo a suitably chosen prime.)<br />
<br />
Problem D.1. Pick two values of a in F_11 = Z/11Z (a not equal to 3), such that the equation y^2 = x^3+ax+1 defines an elliptic curve (i.e., it is smooth). For each such a, determine the number of points #E(F_11) and check that it falls inside the interval described in class. <br />
<br />
Problem D.2. The point P = (0,1) lies on each of these curves. For a =3 (the curve discussed in class) determine the order of P in the elliptic curve group, that is, find the smallest positive integer n such that nP = (infinity) -- the identity element in the group law.<br />
<br />
<!-- '''November 19'''<br />
<br />
Problem A. Using the method discussed in class (which is also the method of problem 4.6 in Stein, which in retrospect I think was too hard to assign with no preparation) find a nontrivial solution with y > 0 to the equation x^2 + 11y^2 = z^2. <br />
Problem B. Let f(X) be the number of solutions of A^2 + B^2 = C^3 such that A^2, B^2, and C^3 are all at most X. Use the heuristic described in class to explain why one might expect f(X) to grow more or less like X^{1/3}.<br />
<br />
Problem C. Show that if A+Bi is the cube of a Gaussian integer, then A^2 + B^2 is a perfect cube.<br />
<br />
Do ONE of problem D1 and D2; D1 is for people who like Sage, D2 is for people who like proving things.<br />
<br />
D1. Use Sage to compute f(X) for X = 1000, 10000, 100000. Does the answer look consistent with the heuristic prediction you made in B? Does f(X)/X^{1/3} appear to be approaching a limit?<br />
D2. We will use problem C to give a lower bound for f(X). Use this to show that there are at least X^{1/3} Gaussian integers with norm at most X that are perfect cubes. From here, show that f(X) > X^{1/3}.<br />
<br />
OPTIONAL: Show that the converse of problem C also holds -- A+Bi has norm a perfect cube if and only if A+Bi is a perfect cube. (You will need unique factorization of Gaussian integers.) Using this fact, prove that f(X) is bounded above by C X^{1/3} for some constant C.<br />
<br />
'''December 8 (note nonstandard due date)'''<br />
<br />
Problem A: Hyperbolas, ellipses, and "magic slopes"<br />
<br />
In the usual analytic geometry, both hyperbolas and ellipses are given by equations of the form<br />
<br />
ax^2 + bxy + cy^2 = d (*)<br />
<br />
<br />
(we always assume d is nonzero.)<br />
<br />
How can we tell whether such an equation describes an ellipse or a<br />
hyperbola? Well, in the geometric setting we all know and love (i.e. in<br />
R^2) a hyperbola has asymptotes and an ellipse does not. What is an <br />
asymptote? You could say it's "a line which doesn't intersect the curve", <br />
but ellipses have such lines too. You might want to say "a line which <br />
doesn't intersect the curve but comes closer and closer to the curve," but <br />
the problem here is that this a) is somewhat imprecise, and b) relies on a <br />
notion of "closer" that is not going to be very clear in Z/pZ, which is <br />
our ultimate goal!<br />
<br />
<br />
So let me state it a different way: let's say the asymptotes of a <br />
hyperbola have slopes m_1 and m_2. These are "magic slopes" in the <br />
following sense: ANY line of slope m_1 (and ditto for m_2) intersects the <br />
hyperbola in at most one point. An ellipse doesn't have "magic slopes" -- <br />
you can convince yourself by drawing pictures that, for any slope m, you <br />
can find a line of slope m striking the ellipse twice.<br />
<br />
<br />
[REMARK: For technical reasons we're going to ignore hyperbolae with a <br />
vertical asymptote, since vertical lines don't exactly have slope.]<br />
<br />
<br />
Now you might know the criterion that, over the real numbers.<br />
<br />
<br />
ax^2 + bxy + cy^2 = d<br />
<br />
<br />
is a hyperbola if b^2 - 4ac > 0 and an ellipse if b^2 - 4ac < 0. (For <br />
instance, x^2 - y^2 = 1 is a hyperbola with magic slopes 1 and -1, and x^2 <br />
+ y^2 = 1 is an ellipse.) <br />
This criterion does not make sense in Z/pZ, where there is no notion of <br />
"greater" or "less."<br />
<br />
<br />
QUESTION A.1.<br />
<br />
For which primes p is it the case that <br />
<br />
x^2 + y^2 = 1<br />
<br />
has magic slopes in Z/pZ? <br />
<br />
QUESTION A.2. For which primes p is it the case that<br />
<br />
<br />
x^2 - y^2 = 1<br />
<br />
<br />
has magic slopes in Z/pZ?<br />
<br />
<br />
QUESTION B: Back in the very first week of this course we studied the set of integers x such that x^2 + 1 is prime. Based on the heuristics we discussed in class, about how many x between 1 and N would you expect to satisfy "x^2 + 1 is prime?"<br />
<br />
<br />
QUESTION C: Let z be a complex number. Then the set of all complex numbers of the form a + bz, with a and b integers, forms a lattice in the complex plane (thought of as R^2.) What is the covolume of this lattice, in terms of z?<br />
<br />
<br />
QUESTION D: Let z be a complex number. Using Minkowski's theorem and the result of question C, show that there exist integers a,b, not both zero, such that Norm(a+bz) is at most (4/pi) Im(z). (Hint: let Omega be the set of all complex numbers of norm at most (4/pi) Im(z).)<br />
<br />
<br />
OPTIONAL EXTRA: Can you give an exact formula for the minimal norm of any nonzero complex number of the form a+bz?<br />
<br />
'''December 12''' Book problems 6.1, 6.2, 6.5, 6.10. For extra credit attempt to do problem 6.9.<br />
<br />
--></div>Andreichttps://www.math.wisc.edu/wiki/index.php?title=Math_567_--_Elementary_Number_Theory&diff=19312Math 567 -- Elementary Number Theory2020-03-27T04:46:57Z<p>Andreic: </p>
<hr />
<div>'''MATH 567<br />
<br />
Elementary Number Theory'''<br />
<br />
MWF 1:20-2:10, Van Vleck B119<br />
<br />
'''Professor:''' [http://www.math.wisc.edu/~andreic/ Andrei Caldararu] (andreic@math.wisc.edu)<br />
<br />
''Office Hours:'' Wednesdays 2:30-3:30, Van Vleck 605. <br />
<br />
'''Grader:''' Yifan Peng, email: peng64@wisc.edu<br />
<br />
''Office Hours:'' TBA.<br />
<br />
Math 567 is a course in elementary number theory, aimed at undergraduates majoring in math or other quantitative disciplines. A general familiarity with abstract algebra at the level of Math 541 will be assumed, but students who haven't taken 541 are welcome to attend if they're willing to play a little catchup. We will be using William Stein's new (and cheap) textbook [http://www.amazon.com/Elementary-Number-Theory-Computational-Undergraduate/dp/0387855246 Elementary Number Theory: Primes, Congruences, and Secrets], which emphasizes computational approaches to the subject. If you don't need a physical copy of the book, [http://www.williamstein.org/ent/ it is available as a free legal .pdf.] We will be using the (free, public-domain) mathematical software [http://www.sagemath.org/ SAGE], developed largely by Stein, as an integral component of our coursework. There is a [http://doc.sagemath.org/pdf/en/tutorial/SageTutorial.pdf useful online tutorial.] You can download SAGE to your own computer or [http://www.sagenb.org use it online].<br />
<br />
Topics include some subset of, but are not limited to: Divisibility, the Euclidean algorithm and the GCD, linear Diophantine equations, prime numbers and uniqueness of factorization. Congruences, Chinese remainder theorem, Fermat's "little" theorem, Wilson's theorem, Euler's theorem and totient function, the RSA cryptosystem, Rabin's encryption scheme, Diffie-Hellman key exchange protocol. Number-theoretic functions, multiplicative functions, Möbius inversion. Primitive roots and indices. Quadratic reciprocity and the Legendre symbol. Perfect numbers, Mersenne primes, Fermat primes. Pythagorean triples, Special cases of Fermat's "last" theorem. Fibonacci numbers. Continued fractions. Distribution of primes, discussion of prime number theorem. Primality testing and factoring algorithms.<br />
<br />
'''Course Policies:''' Homework will be due on Fridays. It can be turned in late only with ''advance'' permission from your grader. It is acceptable to use calculators and computers on homework (indeed, some of it will require a computer) but calculators are not allowed during exams. You are encouraged to work together on homework, but writeups must be done individually.<br />
<br />
Many of the problems in this course will ask you to prove things. I expect proofs to be written in English sentences; the proofs in Stein's book are a good model for the level of verbosity I am looking for. <br />
<br />
'''Grading:''' The grade in Math 567 will be composed of 50% homework, 25% midterm, 25% final. The midterm will be on March 11, in class. The final exam will be on 5/8/2020, 7:45AM-9:45AM in room TBA.<br />
<br />
'''Syllabus:''' <br />
(This may change as we see what pace works well for the course. All section numbers refer to Stein's book.)<br />
<br />
* Jan 22-31: Prime numbers, prime factorizations, Euclidean algorithm and GCD (1.1-1.2)<br />
* Feb 3-7: The integers mod n, Euler's theorem, the phi function (2.1-2.2)<br />
* Feb 10-14: Modular exponentiation, primality testing, and primitive roots (2.4-2.5)<br />
* Feb 17-21: Public-key cryptography and RSA (3.1-3.4)<br />
* Feb 24-28: Rabin's algorithm (not in the book); algebraic numbers <br />
* Mar 2-6: Quadratic reciprocity (4.1-4.4)<br />
* Mar 9, Mar 13: Finite and infinite continued fractions (5.1-5.3)<br />
* Mar 11: ''Midterm exam''<br />
* Mar 23-30: Continued fractions and diophantine approximation (5.4-5.5)<br />
* Apr 1-3: Diophantine equations I: Pell's equation and Lagrange's theorem<br />
* Apr 6-10: More diophantine equations, elliptic curves (6.1)<br />
* Apr 13-17: Applications of elliptic curves (6.2-6.3)<br />
* Apr 20-24: More applications of elliptic curves (6.4)<br />
* Apr 27-May 1: advanced topic TBD: maybe additional discussion of cryptographic techniques?<br />
<br />
'''Video Lectures:'''<br />
Due to the university being transitioned to online teaching only, out lectures after spring break will need to be delivered online. I will post here videos for you to watch for the material we need to cover. Please watch them, and please give me feedback on how they could be improved.<br />
<br />
* '''Lecture for Mar. 23''': watch it [https://mediaspace.wisc.edu/media/Math+567+March+23+Lecture/0_fkhvkxxv here].<br />
* '''Lecture for Mar. 25''': watch it [https://mediaspace.wisc.edu/media/Math+567+March+25+lecture/0_qgkhpm19 here].<br />
* '''Lecture for Mar. 27''': watch it [https://mediaspace.wisc.edu/media/March+27+Math+567+Lecture+Video/0_ajbr2ja0 here].<br />
<br />
'''Homework:'''<br />
Homework is due at the beginning of class on the specified Friday. Typing your homework is not a requirement, but if you don't already know LaTeX I highly recommend that you learn it and use it to typeset your homework. I will sometimes assign extra problems, which I will e-mail to the class list and include here.<br />
<br />
* '''Jan 31''': 1.1, 1.3, 1.5, 1.7 (use SAGE), 1.8, 1.14.<br />
Problem A: Use SAGE to compute the number of x in [1..N] such that x^2 + 1 is prime, for N = 100, N = 1000, and N = 10000. Let f(N) be the number of such x. <br />
<br />
a) Can you formulate a conjecture about the relationship between f(N) and N/log N? <br />
<br />
b) What if x^2 + 1 is replaced with x^2 + 2? Can you explain why x^2 + 2 appears less likely to be prime? (Hint: consider x mod 3.)<br />
<br />
c) Prove that f(N) is at most (1/2)N+1. (Hint: consider x mod 2.) <br />
<br />
d) Give as good an upper bound as you can for f(N).<br />
<br />
Note that, despite the evident regularities you'll observe in this problem, we do not even know whether there are infinitely many primes of the form x^2 + 1! You would become very famous if you proved this.<br />
<br />
* '''Feb 7''': 2.6 (the formulation of numerical evidence should be done by Sage if you've got Sage working, and by calculator if not; you can use an online tool like [http://primes.utm.edu/curios/includes/primetest.php this] to test whether a number is prime.) 2.8,2.9,2.11,2.12,2.14,2.19<br />
<br />
* '''Feb 17''': 2.15, 2.18, 2.23, 2.26, 2.30.<br />
<br />
Problem A: Prove that if n=pq, with p,q prime, then n is not a Carmichael number.<br />
<br />
* '''Feb 21''': Book problems: 3.4, 3.5, 3.6<br />
<br />
Problem A. Prove that there are infinitely many primes p such that 2 is '''not''' a primitive root in Z/pZ. We break this up into steps.<br />
<br />
Problem A.1. Prove that, if x is an element of Z/nZ, then x^2 is not a primitive root.<br />
<br />
Problem A.2. Prove that there are infinitely many primes p such that 2 is a square in Z/pZ. Hint: suppose there are only finitely many such primes p_1, .. p_r, and define N = (p_1 .. p_r)^2 - 2. Where can you go from here...?<br />
<br />
Problem A.3. Give a list of five primes p such that 2 is not a primitive root in Z/pZ (you can use the method of this proof or any other.)<br />
<br />
Problem B. Prove that 24 is the largest integer n such that every element of (Z/nZ)^* is a root of x^2-1.<br />
<br />
* '''Feb 28''': Problem A. Using p = 23 and q=31, show how to encrypt the message x=240 with Rabin's algorithm. Find all possible decryptions of your encrypted message.<br />
<br />
* '''March 23''': 4.1 from the book.<br />
<br />
Problem A. Give a prime factorization of the Gaussian integer 7+9i.<br />
<br />
Problem B. Read the notes from [http://www.math.ucsd.edu/~jverstra/Gaussian1.pdf here]. Note that Z[i] satisfies a ''reduction theorem'': if n and d are Gaussian integers, then there exists integers q and r such that n = qd + r and Norm(r) < Norm(d). But (by contrast with the case of Z) this d may not be unique. In some contexts it is better to be able to choose r uniquely, even if this means letting r have norm greater than Norm(d). <br />
<br />
B.1. When d = 1+2i, show that, for each n in Z[i], there is a '''unique''' pair (q,r) in Z[i] such that n = qd+r and r is contained in the set {0,1,2,3,4}. For instance, i can be written as i(1+2i) + 2, so we say i reduces to 2 mod (1+2i).<br />
(Hint: Suppose q(1+2i)+r = q'(1+2i) + r'. What can we say about (r-r'), and why is this incompatible with both r and r' being in {0,1,2,3,4}?<br />
<br />
B.2. Show that if n is an integer in Z, the reduction of n mod (1+2i) is equal to its reduction mod 5.<br />
<br />
Problem C. Let's try to figure out how to define "phi(d)" for a Gaussian integer d. Suppose S is a set of Gaussian integers such that every n in Z[i] can be written uniquely as qd+r, with q in Z[i] and r in S. (So for instance when d=1+2i, we showed in problem B that we can take S to be {0...4}. It would also be OK to take S to be {1..5} or {0,i,2i,1+i,1+2i}. In fact, it turns out that S has to be a set of size Norm(d) (I might or might not prove this in class; if not, feel free just to accept it.)<br />
<br />
Now define phi(d) to be the number of elements s of S such that s and d are coprime.<br />
<br />
C.1. Compute phi(1+2i) and phi(3).<br />
<br />
C.2. Prove that the value of phi(d) does not depend on the choice of S.<br />
<br />
C.3. Prove that every n in Z[i] which is coprime to 3 satisfies n^phi(3) = 1 mod 3; that is, Euler's theorem holds in this case. (You can prove this by direct computation; of course, if you want, you are welcome to prove that Euler's theorem holds for Z[i] in general, adapting the proof in Stein's book or the one we gave in class.)<br />
<br />
*'''March 30''' <br />
<br />
Problem A. Express 50005 as the sum of two squares.<br />
<br />
In the next two problems we denote by r(n) the number of ways to express n as the sum of two squares (i.e. the number of ordered pairs (a,b) such that a^2 + b^2 = n.) For instance, r(5) = 8: (a,b) = (+/-1, +/-2) or (+/-2, +/-1).<br />
<br />
Problem B. Prove that, for any N, there exists an integer n such that r(n) > N. (I.E., the function r(n) is ''unbounded.'')<br />
<br />
Problem C. If you like Sage, write a short program in Sage to compute r(n) and compute the average of r(n) as n ranges from 1 to 1000. Whether or not you like Sage, make a guess as to how this average would behave if you replaced "1000" by a larger and larger number. (Feel free to ask the Sage-lovers what answer they got in the optional first part of the question.) Can you prove this guess is correct?<br />
<br />
Problem D. We saw in class that the ring Z[sqrt(-5)] doesn't have unique factorization; 6 can be factored as 2*3 or (1+sqrt(-5))(1-sqrt(-5)). In this problem, we will prove that Z[sqrt(-d)] fails to have unique factorization for EVERY odd d >= 5. (Actually it's true for all d >= 5 but to make the proof manageable we'll restrict to the odd case.]<br />
<br />
D.1. Show that (1+sqrt(-d)), (1-sqrt(-d)) and 2 are irreducible in Z[sqrt(-d)].<br />
<br />
D.2. Now give an element of Z[sqrt(-d)] that has two distinct factorizations into irreducibles. (Hint: imitate the example we used for Z[sqrt(-5)].)<br />
''Remark:'' The rings Z[sqrt(d)], where d is positive, are quite different -- here we believe that there are infinitely many with unique factorization, though this conjecture has remained unproved for many decades!<br />
<br />
<!--<br />
*'''November 16'''<br />
<br />
Book problems: 5.3,5.4,5.5<br />
<br />
Problem A: We discussed in class the problem of studying which positive integers are the sum of two squares. In this problem we prove that every element n of (Z/pZ) is the sum of two squares. We argue as follows. Let S be the set of squares in (Z/pZ), and let T be the set of (Z/pZ) consisting of all elements of the form n - x^2, for some x in (Z/pZ).<br />
<br />
A.1. What is the size of S and of T? Use this to show that S and T are not disjoint.<br />
<br />
A.2. Given that S and T are not disjoint, prove that n is the sum of two squares.<br />
<br />
Problem B. Using the continued fraction expansion, find a solution to the Pell equation x^2 - 13 y^2 = 1.<br />
<br />
Problem C. Show that the modified Pell equation x^2 - 7y^2 = -1 has no solutions in integers x,y. (Hint: reduce the equation modulo a suitably chosen prime.)<br />
<br />
Problem D. Stuffy Stirnweiss finished the 1945 season with a batting average of .3085443. Using continued fractions, guess how many at-bats he had. Tony Cuccinello had a batting average of .3084577. Given that he had more than 200 and fewer than 600 at-bats, can you estimate the number of at-bats he had?<br />
<br />
'''December 7'''<br />
<br />
Problem A. When p is a prime congruent to 3 mod 4, prove that ((p-1)/2)! is either 1 or -1 in (Z/pZ). '''OPTIONAL:''' Use sage to compute whether this factorial is 1 or -1 for many primes p. Is there any pattern? Does it seem to be 1 half the time and -1 half the time? Note that I have no idea what the answer to this question is.<br />
<br />
Problem B. Using the continued fraction expansion, find a solution to the Pell equation x^2 - 13 y^2 = 1.<br />
<br />
Problem C. Show that the modified Pell equation x^2 - 7y^2 = -1 has no solutions in integers x,y. (Hint: reduce the equation modulo a suitably chosen prime.)<br />
<br />
Problem D.1. Pick two values of a in F_11 = Z/11Z (a not equal to 3), such that the equation y^2 = x^3+ax+1 defines an elliptic curve (i.e., it is smooth). For each such a, determine the number of points #E(F_11) and check that it falls inside the interval described in class. <br />
<br />
Problem D.2. The point P = (0,1) lies on each of these curves. For a =3 (the curve discussed in class) determine the order of P in the elliptic curve group, that is, find the smallest positive integer n such that nP = (infinity) -- the identity element in the group law.<br />
<br />
<!-- '''November 19'''<br />
<br />
Problem A. Using the method discussed in class (which is also the method of problem 4.6 in Stein, which in retrospect I think was too hard to assign with no preparation) find a nontrivial solution with y > 0 to the equation x^2 + 11y^2 = z^2. <br />
Problem B. Let f(X) be the number of solutions of A^2 + B^2 = C^3 such that A^2, B^2, and C^3 are all at most X. Use the heuristic described in class to explain why one might expect f(X) to grow more or less like X^{1/3}.<br />
<br />
Problem C. Show that if A+Bi is the cube of a Gaussian integer, then A^2 + B^2 is a perfect cube.<br />
<br />
Do ONE of problem D1 and D2; D1 is for people who like Sage, D2 is for people who like proving things.<br />
<br />
D1. Use Sage to compute f(X) for X = 1000, 10000, 100000. Does the answer look consistent with the heuristic prediction you made in B? Does f(X)/X^{1/3} appear to be approaching a limit?<br />
D2. We will use problem C to give a lower bound for f(X). Use this to show that there are at least X^{1/3} Gaussian integers with norm at most X that are perfect cubes. From here, show that f(X) > X^{1/3}.<br />
<br />
OPTIONAL: Show that the converse of problem C also holds -- A+Bi has norm a perfect cube if and only if A+Bi is a perfect cube. (You will need unique factorization of Gaussian integers.) Using this fact, prove that f(X) is bounded above by C X^{1/3} for some constant C.<br />
<br />
'''December 8 (note nonstandard due date)'''<br />
<br />
Problem A: Hyperbolas, ellipses, and "magic slopes"<br />
<br />
In the usual analytic geometry, both hyperbolas and ellipses are given by equations of the form<br />
<br />
ax^2 + bxy + cy^2 = d (*)<br />
<br />
<br />
(we always assume d is nonzero.)<br />
<br />
How can we tell whether such an equation describes an ellipse or a<br />
hyperbola? Well, in the geometric setting we all know and love (i.e. in<br />
R^2) a hyperbola has asymptotes and an ellipse does not. What is an <br />
asymptote? You could say it's "a line which doesn't intersect the curve", <br />
but ellipses have such lines too. You might want to say "a line which <br />
doesn't intersect the curve but comes closer and closer to the curve," but <br />
the problem here is that this a) is somewhat imprecise, and b) relies on a <br />
notion of "closer" that is not going to be very clear in Z/pZ, which is <br />
our ultimate goal!<br />
<br />
<br />
So let me state it a different way: let's say the asymptotes of a <br />
hyperbola have slopes m_1 and m_2. These are "magic slopes" in the <br />
following sense: ANY line of slope m_1 (and ditto for m_2) intersects the <br />
hyperbola in at most one point. An ellipse doesn't have "magic slopes" -- <br />
you can convince yourself by drawing pictures that, for any slope m, you <br />
can find a line of slope m striking the ellipse twice.<br />
<br />
<br />
[REMARK: For technical reasons we're going to ignore hyperbolae with a <br />
vertical asymptote, since vertical lines don't exactly have slope.]<br />
<br />
<br />
Now you might know the criterion that, over the real numbers.<br />
<br />
<br />
ax^2 + bxy + cy^2 = d<br />
<br />
<br />
is a hyperbola if b^2 - 4ac > 0 and an ellipse if b^2 - 4ac < 0. (For <br />
instance, x^2 - y^2 = 1 is a hyperbola with magic slopes 1 and -1, and x^2 <br />
+ y^2 = 1 is an ellipse.) <br />
This criterion does not make sense in Z/pZ, where there is no notion of <br />
"greater" or "less."<br />
<br />
<br />
QUESTION A.1.<br />
<br />
For which primes p is it the case that <br />
<br />
x^2 + y^2 = 1<br />
<br />
has magic slopes in Z/pZ? <br />
<br />
QUESTION A.2. For which primes p is it the case that<br />
<br />
<br />
x^2 - y^2 = 1<br />
<br />
<br />
has magic slopes in Z/pZ?<br />
<br />
<br />
QUESTION B: Back in the very first week of this course we studied the set of integers x such that x^2 + 1 is prime. Based on the heuristics we discussed in class, about how many x between 1 and N would you expect to satisfy "x^2 + 1 is prime?"<br />
<br />
<br />
QUESTION C: Let z be a complex number. Then the set of all complex numbers of the form a + bz, with a and b integers, forms a lattice in the complex plane (thought of as R^2.) What is the covolume of this lattice, in terms of z?<br />
<br />
<br />
QUESTION D: Let z be a complex number. Using Minkowski's theorem and the result of question C, show that there exist integers a,b, not both zero, such that Norm(a+bz) is at most (4/pi) Im(z). (Hint: let Omega be the set of all complex numbers of norm at most (4/pi) Im(z).)<br />
<br />
<br />
OPTIONAL EXTRA: Can you give an exact formula for the minimal norm of any nonzero complex number of the form a+bz?<br />
<br />
'''December 12''' Book problems 6.1, 6.2, 6.5, 6.10. For extra credit attempt to do problem 6.9.<br />
<br />
--></div>Andreichttps://www.math.wisc.edu/wiki/index.php?title=Math_567_--_Elementary_Number_Theory&diff=19303Math 567 -- Elementary Number Theory2020-03-25T00:56:37Z<p>Andreic: </p>
<hr />
<div>'''MATH 567<br />
<br />
Elementary Number Theory'''<br />
<br />
MWF 1:20-2:10, Van Vleck B119<br />
<br />
'''Professor:''' [http://www.math.wisc.edu/~andreic/ Andrei Caldararu] (andreic@math.wisc.edu)<br />
<br />
''Office Hours:'' Wednesdays 2:30-3:30, Van Vleck 605. <br />
<br />
'''Grader:''' Yifan Peng, email: peng64@wisc.edu<br />
<br />
''Office Hours:'' TBA.<br />
<br />
Math 567 is a course in elementary number theory, aimed at undergraduates majoring in math or other quantitative disciplines. A general familiarity with abstract algebra at the level of Math 541 will be assumed, but students who haven't taken 541 are welcome to attend if they're willing to play a little catchup. We will be using William Stein's new (and cheap) textbook [http://www.amazon.com/Elementary-Number-Theory-Computational-Undergraduate/dp/0387855246 Elementary Number Theory: Primes, Congruences, and Secrets], which emphasizes computational approaches to the subject. If you don't need a physical copy of the book, [http://www.williamstein.org/ent/ it is available as a free legal .pdf.] We will be using the (free, public-domain) mathematical software [http://www.sagemath.org/ SAGE], developed largely by Stein, as an integral component of our coursework. There is a [http://doc.sagemath.org/pdf/en/tutorial/SageTutorial.pdf useful online tutorial.] You can download SAGE to your own computer or [http://www.sagenb.org use it online].<br />
<br />
Topics include some subset of, but are not limited to: Divisibility, the Euclidean algorithm and the GCD, linear Diophantine equations, prime numbers and uniqueness of factorization. Congruences, Chinese remainder theorem, Fermat's "little" theorem, Wilson's theorem, Euler's theorem and totient function, the RSA cryptosystem, Rabin's encryption scheme, Diffie-Hellman key exchange protocol. Number-theoretic functions, multiplicative functions, Möbius inversion. Primitive roots and indices. Quadratic reciprocity and the Legendre symbol. Perfect numbers, Mersenne primes, Fermat primes. Pythagorean triples, Special cases of Fermat's "last" theorem. Fibonacci numbers. Continued fractions. Distribution of primes, discussion of prime number theorem. Primality testing and factoring algorithms.<br />
<br />
'''Course Policies:''' Homework will be due on Fridays. It can be turned in late only with ''advance'' permission from your grader. It is acceptable to use calculators and computers on homework (indeed, some of it will require a computer) but calculators are not allowed during exams. You are encouraged to work together on homework, but writeups must be done individually.<br />
<br />
Many of the problems in this course will ask you to prove things. I expect proofs to be written in English sentences; the proofs in Stein's book are a good model for the level of verbosity I am looking for. <br />
<br />
'''Grading:''' The grade in Math 567 will be composed of 50% homework, 25% midterm, 25% final. The midterm will be on March 11, in class. The final exam will be on 5/8/2020, 7:45AM-9:45AM in room TBA.<br />
<br />
'''Syllabus:''' <br />
(This may change as we see what pace works well for the course. All section numbers refer to Stein's book.)<br />
<br />
* Jan 22-31: Prime numbers, prime factorizations, Euclidean algorithm and GCD (1.1-1.2)<br />
* Feb 3-7: The integers mod n, Euler's theorem, the phi function (2.1-2.2)<br />
* Feb 10-14: Modular exponentiation, primality testing, and primitive roots (2.4-2.5)<br />
* Feb 17-21: Public-key cryptography and RSA (3.1-3.4)<br />
* Feb 24-28: Rabin's algorithm (not in the book); algebraic numbers <br />
* Mar 2-6: Quadratic reciprocity (4.1-4.4)<br />
* Mar 9, Mar 13: Finite and infinite continued fractions (5.1-5.3)<br />
* Mar 11: ''Midterm exam''<br />
* Mar 23-30: Continued fractions and diophantine approximation (5.4-5.5)<br />
* Apr 1-3: Diophantine equations I: Pell's equation and Lagrange's theorem<br />
* Apr 6-10: More diophantine equations, elliptic curves (6.1)<br />
* Apr 13-17: Applications of elliptic curves (6.2-6.3)<br />
* Apr 20-24: More applications of elliptic curves (6.4)<br />
* Apr 27-May 1: advanced topic TBD: maybe additional discussion of cryptographic techniques?<br />
<br />
'''Video Lectures:'''<br />
Due to the university being transitioned to online teaching only, out lectures after spring break will need to be delivered online. I will post here videos for you to watch for the material we need to cover. Please watch them, and please give me feedback on how they could be improved.<br />
<br />
* '''Lecture for Mar. 23''': watch it [https://mediaspace.wisc.edu/media/Math+567+March+23+Lecture/0_fkhvkxxv here].<br />
* '''Lecture for Mar. 25''': watch it [https://mediaspace.wisc.edu/media/Math+567+March+25+lecture/0_qgkhpm19 here].<br />
<br />
'''Homework:'''<br />
Homework is due at the beginning of class on the specified Friday. Typing your homework is not a requirement, but if you don't already know LaTeX I highly recommend that you learn it and use it to typeset your homework. I will sometimes assign extra problems, which I will e-mail to the class list and include here.<br />
<br />
* '''Jan 31''': 1.1, 1.3, 1.5, 1.7 (use SAGE), 1.8, 1.14.<br />
Problem A: Use SAGE to compute the number of x in [1..N] such that x^2 + 1 is prime, for N = 100, N = 1000, and N = 10000. Let f(N) be the number of such x. <br />
<br />
a) Can you formulate a conjecture about the relationship between f(N) and N/log N? <br />
<br />
b) What if x^2 + 1 is replaced with x^2 + 2? Can you explain why x^2 + 2 appears less likely to be prime? (Hint: consider x mod 3.)<br />
<br />
c) Prove that f(N) is at most (1/2)N+1. (Hint: consider x mod 2.) <br />
<br />
d) Give as good an upper bound as you can for f(N).<br />
<br />
Note that, despite the evident regularities you'll observe in this problem, we do not even know whether there are infinitely many primes of the form x^2 + 1! You would become very famous if you proved this.<br />
<br />
* '''Feb 7''': 2.6 (the formulation of numerical evidence should be done by Sage if you've got Sage working, and by calculator if not; you can use an online tool like [http://primes.utm.edu/curios/includes/primetest.php this] to test whether a number is prime.) 2.8,2.9,2.11,2.12,2.14,2.19<br />
<br />
* '''Feb 17''': 2.15, 2.18, 2.23, 2.26, 2.30.<br />
<br />
Problem A: Prove that if n=pq, with p,q prime, then n is not a Carmichael number.<br />
<br />
* '''Feb 21''': Book problems: 3.4, 3.5, 3.6<br />
<br />
Problem A. Prove that there are infinitely many primes p such that 2 is '''not''' a primitive root in Z/pZ. We break this up into steps.<br />
<br />
Problem A.1. Prove that, if x is an element of Z/nZ, then x^2 is not a primitive root.<br />
<br />
Problem A.2. Prove that there are infinitely many primes p such that 2 is a square in Z/pZ. Hint: suppose there are only finitely many such primes p_1, .. p_r, and define N = (p_1 .. p_r)^2 - 2. Where can you go from here...?<br />
<br />
Problem A.3. Give a list of five primes p such that 2 is not a primitive root in Z/pZ (you can use the method of this proof or any other.)<br />
<br />
Problem B. Prove that 24 is the largest integer n such that every element of (Z/nZ)^* is a root of x^2-1.<br />
<br />
* '''Feb 28''': Problem A. Using p = 23 and q=31, show how to encrypt the message x=240 with Rabin's algorithm. Find all possible decryptions of your encrypted message.<br />
<br />
* '''March 23''': 4.1 from the book.<br />
<br />
Problem A. Give a prime factorization of the Gaussian integer 7+9i.<br />
<br />
Problem B. Read the notes from [http://www.math.ucsd.edu/~jverstra/Gaussian1.pdf here]. Note that Z[i] satisfies a ''reduction theorem'': if n and d are Gaussian integers, then there exists integers q and r such that n = qd + r and Norm(r) < Norm(d). But (by contrast with the case of Z) this d may not be unique. In some contexts it is better to be able to choose r uniquely, even if this means letting r have norm greater than Norm(d). <br />
<br />
B.1. When d = 1+2i, show that, for each n in Z[i], there is a '''unique''' pair (q,r) in Z[i] such that n = qd+r and r is contained in the set {0,1,2,3,4}. For instance, i can be written as i(1+2i) + 2, so we say i reduces to 2 mod (1+2i).<br />
(Hint: Suppose q(1+2i)+r = q'(1+2i) + r'. What can we say about (r-r'), and why is this incompatible with both r and r' being in {0,1,2,3,4}?<br />
<br />
B.2. Show that if n is an integer in Z, the reduction of n mod (1+2i) is equal to its reduction mod 5.<br />
<br />
Problem C. Let's try to figure out how to define "phi(d)" for a Gaussian integer d. Suppose S is a set of Gaussian integers such that every n in Z[i] can be written uniquely as qd+r, with q in Z[i] and r in S. (So for instance when d=1+2i, we showed in problem B that we can take S to be {0...4}. It would also be OK to take S to be {1..5} or {0,i,2i,1+i,1+2i}. In fact, it turns out that S has to be a set of size Norm(d) (I might or might not prove this in class; if not, feel free just to accept it.)<br />
<br />
Now define phi(d) to be the number of elements s of S such that s and d are coprime.<br />
<br />
C.1. Compute phi(1+2i) and phi(3).<br />
<br />
C.2. Prove that the value of phi(d) does not depend on the choice of S.<br />
<br />
C.3. Prove that every n in Z[i] which is coprime to 3 satisfies n^phi(3) = 1 mod 3; that is, Euler's theorem holds in this case. (You can prove this by direct computation; of course, if you want, you are welcome to prove that Euler's theorem holds for Z[i] in general, adapting the proof in Stein's book or the one we gave in class.)<br />
<br />
*'''March 30''' <br />
<br />
Problem A. Express 50005 as the sum of two squares.<br />
<br />
In the next two problems we denote by r(n) the number of ways to express n as the sum of two squares (i.e. the number of ordered pairs (a,b) such that a^2 + b^2 = n.) For instance, r(5) = 8: (a,b) = (+/-1, +/-2) or (+/-2, +/-1).<br />
<br />
Problem B. Prove that, for any N, there exists an integer n such that r(n) > N. (I.E., the function r(n) is ''unbounded.'')<br />
<br />
Problem C. If you like Sage, write a short program in Sage to compute r(n) and compute the average of r(n) as n ranges from 1 to 1000. Whether or not you like Sage, make a guess as to how this average would behave if you replaced "1000" by a larger and larger number. (Feel free to ask the Sage-lovers what answer they got in the optional first part of the question.) Can you prove this guess is correct?<br />
<br />
Problem D. We saw in class that the ring Z[sqrt(-5)] doesn't have unique factorization; 6 can be factored as 2*3 or (1+sqrt(-5))(1-sqrt(-5)). In this problem, we will prove that Z[sqrt(-d)] fails to have unique factorization for EVERY odd d >= 5. (Actually it's true for all d >= 5 but to make the proof manageable we'll restrict to the odd case.]<br />
<br />
D.1. Show that (1+sqrt(-d)), (1-sqrt(-d)) and 2 are irreducible in Z[sqrt(-d)].<br />
<br />
D.2. Now give an element of Z[sqrt(-d)] that has two distinct factorizations into irreducibles. (Hint: imitate the example we used for Z[sqrt(-5)].)<br />
''Remark:'' The rings Z[sqrt(d)], where d is positive, are quite different -- here we believe that there are infinitely many with unique factorization, though this conjecture has remained unproved for many decades!<br />
<br />
<!--<br />
*'''November 16'''<br />
<br />
Book problems: 5.3,5.4,5.5<br />
<br />
Problem A: We discussed in class the problem of studying which positive integers are the sum of two squares. In this problem we prove that every element n of (Z/pZ) is the sum of two squares. We argue as follows. Let S be the set of squares in (Z/pZ), and let T be the set of (Z/pZ) consisting of all elements of the form n - x^2, for some x in (Z/pZ).<br />
<br />
A.1. What is the size of S and of T? Use this to show that S and T are not disjoint.<br />
<br />
A.2. Given that S and T are not disjoint, prove that n is the sum of two squares.<br />
<br />
Problem B. Using the continued fraction expansion, find a solution to the Pell equation x^2 - 13 y^2 = 1.<br />
<br />
Problem C. Show that the modified Pell equation x^2 - 7y^2 = -1 has no solutions in integers x,y. (Hint: reduce the equation modulo a suitably chosen prime.)<br />
<br />
Problem D. Stuffy Stirnweiss finished the 1945 season with a batting average of .3085443. Using continued fractions, guess how many at-bats he had. Tony Cuccinello had a batting average of .3084577. Given that he had more than 200 and fewer than 600 at-bats, can you estimate the number of at-bats he had?<br />
<br />
'''December 7'''<br />
<br />
Problem A. When p is a prime congruent to 3 mod 4, prove that ((p-1)/2)! is either 1 or -1 in (Z/pZ). '''OPTIONAL:''' Use sage to compute whether this factorial is 1 or -1 for many primes p. Is there any pattern? Does it seem to be 1 half the time and -1 half the time? Note that I have no idea what the answer to this question is.<br />
<br />
Problem B. Using the continued fraction expansion, find a solution to the Pell equation x^2 - 13 y^2 = 1.<br />
<br />
Problem C. Show that the modified Pell equation x^2 - 7y^2 = -1 has no solutions in integers x,y. (Hint: reduce the equation modulo a suitably chosen prime.)<br />
<br />
Problem D.1. Pick two values of a in F_11 = Z/11Z (a not equal to 3), such that the equation y^2 = x^3+ax+1 defines an elliptic curve (i.e., it is smooth). For each such a, determine the number of points #E(F_11) and check that it falls inside the interval described in class. <br />
<br />
Problem D.2. The point P = (0,1) lies on each of these curves. For a =3 (the curve discussed in class) determine the order of P in the elliptic curve group, that is, find the smallest positive integer n such that nP = (infinity) -- the identity element in the group law.<br />
<br />
<!-- '''November 19'''<br />
<br />
Problem A. Using the method discussed in class (which is also the method of problem 4.6 in Stein, which in retrospect I think was too hard to assign with no preparation) find a nontrivial solution with y > 0 to the equation x^2 + 11y^2 = z^2. <br />
Problem B. Let f(X) be the number of solutions of A^2 + B^2 = C^3 such that A^2, B^2, and C^3 are all at most X. Use the heuristic described in class to explain why one might expect f(X) to grow more or less like X^{1/3}.<br />
<br />
Problem C. Show that if A+Bi is the cube of a Gaussian integer, then A^2 + B^2 is a perfect cube.<br />
<br />
Do ONE of problem D1 and D2; D1 is for people who like Sage, D2 is for people who like proving things.<br />
<br />
D1. Use Sage to compute f(X) for X = 1000, 10000, 100000. Does the answer look consistent with the heuristic prediction you made in B? Does f(X)/X^{1/3} appear to be approaching a limit?<br />
D2. We will use problem C to give a lower bound for f(X). Use this to show that there are at least X^{1/3} Gaussian integers with norm at most X that are perfect cubes. From here, show that f(X) > X^{1/3}.<br />
<br />
OPTIONAL: Show that the converse of problem C also holds -- A+Bi has norm a perfect cube if and only if A+Bi is a perfect cube. (You will need unique factorization of Gaussian integers.) Using this fact, prove that f(X) is bounded above by C X^{1/3} for some constant C.<br />
<br />
'''December 8 (note nonstandard due date)'''<br />
<br />
Problem A: Hyperbolas, ellipses, and "magic slopes"<br />
<br />
In the usual analytic geometry, both hyperbolas and ellipses are given by equations of the form<br />
<br />
ax^2 + bxy + cy^2 = d (*)<br />
<br />
<br />
(we always assume d is nonzero.)<br />
<br />
How can we tell whether such an equation describes an ellipse or a<br />
hyperbola? Well, in the geometric setting we all know and love (i.e. in<br />
R^2) a hyperbola has asymptotes and an ellipse does not. What is an <br />
asymptote? You could say it's "a line which doesn't intersect the curve", <br />
but ellipses have such lines too. You might want to say "a line which <br />
doesn't intersect the curve but comes closer and closer to the curve," but <br />
the problem here is that this a) is somewhat imprecise, and b) relies on a <br />
notion of "closer" that is not going to be very clear in Z/pZ, which is <br />
our ultimate goal!<br />
<br />
<br />
So let me state it a different way: let's say the asymptotes of a <br />
hyperbola have slopes m_1 and m_2. These are "magic slopes" in the <br />
following sense: ANY line of slope m_1 (and ditto for m_2) intersects the <br />
hyperbola in at most one point. An ellipse doesn't have "magic slopes" -- <br />
you can convince yourself by drawing pictures that, for any slope m, you <br />
can find a line of slope m striking the ellipse twice.<br />
<br />
<br />
[REMARK: For technical reasons we're going to ignore hyperbolae with a <br />
vertical asymptote, since vertical lines don't exactly have slope.]<br />
<br />
<br />
Now you might know the criterion that, over the real numbers.<br />
<br />
<br />
ax^2 + bxy + cy^2 = d<br />
<br />
<br />
is a hyperbola if b^2 - 4ac > 0 and an ellipse if b^2 - 4ac < 0. (For <br />
instance, x^2 - y^2 = 1 is a hyperbola with magic slopes 1 and -1, and x^2 <br />
+ y^2 = 1 is an ellipse.) <br />
This criterion does not make sense in Z/pZ, where there is no notion of <br />
"greater" or "less."<br />
<br />
<br />
QUESTION A.1.<br />
<br />
For which primes p is it the case that <br />
<br />
x^2 + y^2 = 1<br />
<br />
has magic slopes in Z/pZ? <br />
<br />
QUESTION A.2. For which primes p is it the case that<br />
<br />
<br />
x^2 - y^2 = 1<br />
<br />
<br />
has magic slopes in Z/pZ?<br />
<br />
<br />
QUESTION B: Back in the very first week of this course we studied the set of integers x such that x^2 + 1 is prime. Based on the heuristics we discussed in class, about how many x between 1 and N would you expect to satisfy "x^2 + 1 is prime?"<br />
<br />
<br />
QUESTION C: Let z be a complex number. Then the set of all complex numbers of the form a + bz, with a and b integers, forms a lattice in the complex plane (thought of as R^2.) What is the covolume of this lattice, in terms of z?<br />
<br />
<br />
QUESTION D: Let z be a complex number. Using Minkowski's theorem and the result of question C, show that there exist integers a,b, not both zero, such that Norm(a+bz) is at most (4/pi) Im(z). (Hint: let Omega be the set of all complex numbers of norm at most (4/pi) Im(z).)<br />
<br />
<br />
OPTIONAL EXTRA: Can you give an exact formula for the minimal norm of any nonzero complex number of the form a+bz?<br />
<br />
'''December 12''' Book problems 6.1, 6.2, 6.5, 6.10. For extra credit attempt to do problem 6.9.<br />
<br />
--></div>Andreichttps://www.math.wisc.edu/wiki/index.php?title=Math_567_--_Elementary_Number_Theory&diff=19294Math 567 -- Elementary Number Theory2020-03-23T16:05:07Z<p>Andreic: </p>
<hr />
<div>'''MATH 567<br />
<br />
Elementary Number Theory'''<br />
<br />
MWF 1:20-2:10, Van Vleck B119<br />
<br />
'''Professor:''' [http://www.math.wisc.edu/~andreic/ Andrei Caldararu] (andreic@math.wisc.edu)<br />
<br />
''Office Hours:'' Wednesdays 2:30-3:30, Van Vleck 605. <br />
<br />
'''Grader:''' Yifan Peng, email: peng64@wisc.edu<br />
<br />
''Office Hours:'' TBA.<br />
<br />
Math 567 is a course in elementary number theory, aimed at undergraduates majoring in math or other quantitative disciplines. A general familiarity with abstract algebra at the level of Math 541 will be assumed, but students who haven't taken 541 are welcome to attend if they're willing to play a little catchup. We will be using William Stein's new (and cheap) textbook [http://www.amazon.com/Elementary-Number-Theory-Computational-Undergraduate/dp/0387855246 Elementary Number Theory: Primes, Congruences, and Secrets], which emphasizes computational approaches to the subject. If you don't need a physical copy of the book, [http://www.williamstein.org/ent/ it is available as a free legal .pdf.] We will be using the (free, public-domain) mathematical software [http://www.sagemath.org/ SAGE], developed largely by Stein, as an integral component of our coursework. There is a [http://doc.sagemath.org/pdf/en/tutorial/SageTutorial.pdf useful online tutorial.] You can download SAGE to your own computer or [http://www.sagenb.org use it online].<br />
<br />
Topics include some subset of, but are not limited to: Divisibility, the Euclidean algorithm and the GCD, linear Diophantine equations, prime numbers and uniqueness of factorization. Congruences, Chinese remainder theorem, Fermat's "little" theorem, Wilson's theorem, Euler's theorem and totient function, the RSA cryptosystem, Rabin's encryption scheme, Diffie-Hellman key exchange protocol. Number-theoretic functions, multiplicative functions, Möbius inversion. Primitive roots and indices. Quadratic reciprocity and the Legendre symbol. Perfect numbers, Mersenne primes, Fermat primes. Pythagorean triples, Special cases of Fermat's "last" theorem. Fibonacci numbers. Continued fractions. Distribution of primes, discussion of prime number theorem. Primality testing and factoring algorithms.<br />
<br />
'''Course Policies:''' Homework will be due on Fridays. It can be turned in late only with ''advance'' permission from your grader. It is acceptable to use calculators and computers on homework (indeed, some of it will require a computer) but calculators are not allowed during exams. You are encouraged to work together on homework, but writeups must be done individually.<br />
<br />
Many of the problems in this course will ask you to prove things. I expect proofs to be written in English sentences; the proofs in Stein's book are a good model for the level of verbosity I am looking for. <br />
<br />
'''Grading:''' The grade in Math 567 will be composed of 50% homework, 25% midterm, 25% final. The midterm will be on March 11, in class. The final exam will be on 5/8/2020, 7:45AM-9:45AM in room TBA.<br />
<br />
'''Syllabus:''' <br />
(This may change as we see what pace works well for the course. All section numbers refer to Stein's book.)<br />
<br />
* Jan 22-31: Prime numbers, prime factorizations, Euclidean algorithm and GCD (1.1-1.2)<br />
* Feb 3-7: The integers mod n, Euler's theorem, the phi function (2.1-2.2)<br />
* Feb 10-14: Modular exponentiation, primality testing, and primitive roots (2.4-2.5)<br />
* Feb 17-21: Public-key cryptography and RSA (3.1-3.4)<br />
* Feb 24-28: Rabin's algorithm (not in the book); algebraic numbers <br />
* Mar 2-6: Quadratic reciprocity (4.1-4.4)<br />
* Mar 9, Mar 13: Finite and infinite continued fractions (5.1-5.3)<br />
* Mar 11: ''Midterm exam''<br />
* Mar 23-30: Continued fractions and diophantine approximation (5.4-5.5)<br />
* Apr 1-3: Diophantine equations I: Pell's equation and Lagrange's theorem<br />
* Apr 6-10: More diophantine equations, elliptic curves (6.1)<br />
* Apr 13-17: Applications of elliptic curves (6.2-6.3)<br />
* Apr 20-24: More applications of elliptic curves (6.4)<br />
* Apr 27-May 1: advanced topic TBD: maybe additional discussion of cryptographic techniques?<br />
<br />
'''Video Lectures:'''<br />
Due to the university being transitioned to online teaching only, out lectures after spring break will need to be delivered online. I will post here videos for you to watch for the material we need to cover. Please watch them, and please give me feedback on how they could be improved.<br />
<br />
* '''Lecture for Mar. 23''': watch it [https://mediaspace.wisc.edu/media/Math+567+March+23+Lecture/0_fkhvkxxv here].<br />
<br />
'''Homework:'''<br />
Homework is due at the beginning of class on the specified Friday. Typing your homework is not a requirement, but if you don't already know LaTeX I highly recommend that you learn it and use it to typeset your homework. I will sometimes assign extra problems, which I will e-mail to the class list and include here.<br />
<br />
* '''Jan 31''': 1.1, 1.3, 1.5, 1.7 (use SAGE), 1.8, 1.14.<br />
Problem A: Use SAGE to compute the number of x in [1..N] such that x^2 + 1 is prime, for N = 100, N = 1000, and N = 10000. Let f(N) be the number of such x. <br />
<br />
a) Can you formulate a conjecture about the relationship between f(N) and N/log N? <br />
<br />
b) What if x^2 + 1 is replaced with x^2 + 2? Can you explain why x^2 + 2 appears less likely to be prime? (Hint: consider x mod 3.)<br />
<br />
c) Prove that f(N) is at most (1/2)N+1. (Hint: consider x mod 2.) <br />
<br />
d) Give as good an upper bound as you can for f(N).<br />
<br />
Note that, despite the evident regularities you'll observe in this problem, we do not even know whether there are infinitely many primes of the form x^2 + 1! You would become very famous if you proved this.<br />
<br />
* '''Feb 7''': 2.6 (the formulation of numerical evidence should be done by Sage if you've got Sage working, and by calculator if not; you can use an online tool like [http://primes.utm.edu/curios/includes/primetest.php this] to test whether a number is prime.) 2.8,2.9,2.11,2.12,2.14,2.19<br />
<br />
* '''Feb 17''': 2.15, 2.18, 2.23, 2.26, 2.30.<br />
<br />
Problem A: Prove that if n=pq, with p,q prime, then n is not a Carmichael number.<br />
<br />
* '''Feb 21''': Book problems: 3.4, 3.5, 3.6<br />
<br />
Problem A. Prove that there are infinitely many primes p such that 2 is '''not''' a primitive root in Z/pZ. We break this up into steps.<br />
<br />
Problem A.1. Prove that, if x is an element of Z/nZ, then x^2 is not a primitive root.<br />
<br />
Problem A.2. Prove that there are infinitely many primes p such that 2 is a square in Z/pZ. Hint: suppose there are only finitely many such primes p_1, .. p_r, and define N = (p_1 .. p_r)^2 - 2. Where can you go from here...?<br />
<br />
Problem A.3. Give a list of five primes p such that 2 is not a primitive root in Z/pZ (you can use the method of this proof or any other.)<br />
<br />
Problem B. Prove that 24 is the largest integer n such that every element of (Z/nZ)^* is a root of x^2-1.<br />
<br />
* '''Feb 28''': Problem A. Using p = 23 and q=31, show how to encrypt the message x=240 with Rabin's algorithm. Find all possible decryptions of your encrypted message.<br />
<br />
* '''March 23''': 4.1 from the book.<br />
<br />
Problem A. Give a prime factorization of the Gaussian integer 7+9i.<br />
<br />
Problem B. Read the notes from [http://www.math.ucsd.edu/~jverstra/Gaussian1.pdf here]. Note that Z[i] satisfies a ''reduction theorem'': if n and d are Gaussian integers, then there exists integers q and r such that n = qd + r and Norm(r) < Norm(d). But (by contrast with the case of Z) this d may not be unique. In some contexts it is better to be able to choose r uniquely, even if this means letting r have norm greater than Norm(d). <br />
<br />
B.1. When d = 1+2i, show that, for each n in Z[i], there is a '''unique''' pair (q,r) in Z[i] such that n = qd+r and r is contained in the set {0,1,2,3,4}. For instance, i can be written as i(1+2i) + 2, so we say i reduces to 2 mod (1+2i).<br />
(Hint: Suppose q(1+2i)+r = q'(1+2i) + r'. What can we say about (r-r'), and why is this incompatible with both r and r' being in {0,1,2,3,4}?<br />
<br />
B.2. Show that if n is an integer in Z, the reduction of n mod (1+2i) is equal to its reduction mod 5.<br />
<br />
Problem C. Let's try to figure out how to define "phi(d)" for a Gaussian integer d. Suppose S is a set of Gaussian integers such that every n in Z[i] can be written uniquely as qd+r, with q in Z[i] and r in S. (So for instance when d=1+2i, we showed in problem B that we can take S to be {0...4}. It would also be OK to take S to be {1..5} or {0,i,2i,1+i,1+2i}. In fact, it turns out that S has to be a set of size Norm(d) (I might or might not prove this in class; if not, feel free just to accept it.)<br />
<br />
Now define phi(d) to be the number of elements s of S such that s and d are coprime.<br />
<br />
C.1. Compute phi(1+2i) and phi(3).<br />
<br />
C.2. Prove that the value of phi(d) does not depend on the choice of S.<br />
<br />
C.3. Prove that every n in Z[i] which is coprime to 3 satisfies n^phi(3) = 1 mod 3; that is, Euler's theorem holds in this case. (You can prove this by direct computation; of course, if you want, you are welcome to prove that Euler's theorem holds for Z[i] in general, adapting the proof in Stein's book or the one we gave in class.)<br />
<br />
*'''March 30''' <br />
<br />
Problem A. Express 50005 as the sum of two squares.<br />
<br />
In the next two problems we denote by r(n) the number of ways to express n as the sum of two squares (i.e. the number of ordered pairs (a,b) such that a^2 + b^2 = n.) For instance, r(5) = 8: (a,b) = (+/-1, +/-2) or (+/-2, +/-1).<br />
<br />
Problem B. Prove that, for any N, there exists an integer n such that r(n) > N. (I.E., the function r(n) is ''unbounded.'')<br />
<br />
Problem C. If you like Sage, write a short program in Sage to compute r(n) and compute the average of r(n) as n ranges from 1 to 1000. Whether or not you like Sage, make a guess as to how this average would behave if you replaced "1000" by a larger and larger number. (Feel free to ask the Sage-lovers what answer they got in the optional first part of the question.) Can you prove this guess is correct?<br />
<br />
Problem D. We saw in class that the ring Z[sqrt(-5)] doesn't have unique factorization; 6 can be factored as 2*3 or (1+sqrt(-5))(1-sqrt(-5)). In this problem, we will prove that Z[sqrt(-d)] fails to have unique factorization for EVERY odd d >= 5. (Actually it's true for all d >= 5 but to make the proof manageable we'll restrict to the odd case.]<br />
<br />
D.1. Show that (1+sqrt(-d)), (1-sqrt(-d)) and 2 are irreducible in Z[sqrt(-d)].<br />
<br />
D.2. Now give an element of Z[sqrt(-d)] that has two distinct factorizations into irreducibles. (Hint: imitate the example we used for Z[sqrt(-5)].)<br />
''Remark:'' The rings Z[sqrt(d)], where d is positive, are quite different -- here we believe that there are infinitely many with unique factorization, though this conjecture has remained unproved for many decades!<br />
<br />
<!--<br />
*'''November 16'''<br />
<br />
Book problems: 5.3,5.4,5.5<br />
<br />
Problem A: We discussed in class the problem of studying which positive integers are the sum of two squares. In this problem we prove that every element n of (Z/pZ) is the sum of two squares. We argue as follows. Let S be the set of squares in (Z/pZ), and let T be the set of (Z/pZ) consisting of all elements of the form n - x^2, for some x in (Z/pZ).<br />
<br />
A.1. What is the size of S and of T? Use this to show that S and T are not disjoint.<br />
<br />
A.2. Given that S and T are not disjoint, prove that n is the sum of two squares.<br />
<br />
Problem B. Using the continued fraction expansion, find a solution to the Pell equation x^2 - 13 y^2 = 1.<br />
<br />
Problem C. Show that the modified Pell equation x^2 - 7y^2 = -1 has no solutions in integers x,y. (Hint: reduce the equation modulo a suitably chosen prime.)<br />
<br />
Problem D. Stuffy Stirnweiss finished the 1945 season with a batting average of .3085443. Using continued fractions, guess how many at-bats he had. Tony Cuccinello had a batting average of .3084577. Given that he had more than 200 and fewer than 600 at-bats, can you estimate the number of at-bats he had?<br />
<br />
'''December 7'''<br />
<br />
Problem A. When p is a prime congruent to 3 mod 4, prove that ((p-1)/2)! is either 1 or -1 in (Z/pZ). '''OPTIONAL:''' Use sage to compute whether this factorial is 1 or -1 for many primes p. Is there any pattern? Does it seem to be 1 half the time and -1 half the time? Note that I have no idea what the answer to this question is.<br />
<br />
Problem B. Using the continued fraction expansion, find a solution to the Pell equation x^2 - 13 y^2 = 1.<br />
<br />
Problem C. Show that the modified Pell equation x^2 - 7y^2 = -1 has no solutions in integers x,y. (Hint: reduce the equation modulo a suitably chosen prime.)<br />
<br />
Problem D.1. Pick two values of a in F_11 = Z/11Z (a not equal to 3), such that the equation y^2 = x^3+ax+1 defines an elliptic curve (i.e., it is smooth). For each such a, determine the number of points #E(F_11) and check that it falls inside the interval described in class. <br />
<br />
Problem D.2. The point P = (0,1) lies on each of these curves. For a =3 (the curve discussed in class) determine the order of P in the elliptic curve group, that is, find the smallest positive integer n such that nP = (infinity) -- the identity element in the group law.<br />
<br />
<!-- '''November 19'''<br />
<br />
Problem A. Using the method discussed in class (which is also the method of problem 4.6 in Stein, which in retrospect I think was too hard to assign with no preparation) find a nontrivial solution with y > 0 to the equation x^2 + 11y^2 = z^2. <br />
Problem B. Let f(X) be the number of solutions of A^2 + B^2 = C^3 such that A^2, B^2, and C^3 are all at most X. Use the heuristic described in class to explain why one might expect f(X) to grow more or less like X^{1/3}.<br />
<br />
Problem C. Show that if A+Bi is the cube of a Gaussian integer, then A^2 + B^2 is a perfect cube.<br />
<br />
Do ONE of problem D1 and D2; D1 is for people who like Sage, D2 is for people who like proving things.<br />
<br />
D1. Use Sage to compute f(X) for X = 1000, 10000, 100000. Does the answer look consistent with the heuristic prediction you made in B? Does f(X)/X^{1/3} appear to be approaching a limit?<br />
D2. We will use problem C to give a lower bound for f(X). Use this to show that there are at least X^{1/3} Gaussian integers with norm at most X that are perfect cubes. From here, show that f(X) > X^{1/3}.<br />
<br />
OPTIONAL: Show that the converse of problem C also holds -- A+Bi has norm a perfect cube if and only if A+Bi is a perfect cube. (You will need unique factorization of Gaussian integers.) Using this fact, prove that f(X) is bounded above by C X^{1/3} for some constant C.<br />
<br />
'''December 8 (note nonstandard due date)'''<br />
<br />
Problem A: Hyperbolas, ellipses, and "magic slopes"<br />
<br />
In the usual analytic geometry, both hyperbolas and ellipses are given by equations of the form<br />
<br />
ax^2 + bxy + cy^2 = d (*)<br />
<br />
<br />
(we always assume d is nonzero.)<br />
<br />
How can we tell whether such an equation describes an ellipse or a<br />
hyperbola? Well, in the geometric setting we all know and love (i.e. in<br />
R^2) a hyperbola has asymptotes and an ellipse does not. What is an <br />
asymptote? You could say it's "a line which doesn't intersect the curve", <br />
but ellipses have such lines too. You might want to say "a line which <br />
doesn't intersect the curve but comes closer and closer to the curve," but <br />
the problem here is that this a) is somewhat imprecise, and b) relies on a <br />
notion of "closer" that is not going to be very clear in Z/pZ, which is <br />
our ultimate goal!<br />
<br />
<br />
So let me state it a different way: let's say the asymptotes of a <br />
hyperbola have slopes m_1 and m_2. These are "magic slopes" in the <br />
following sense: ANY line of slope m_1 (and ditto for m_2) intersects the <br />
hyperbola in at most one point. An ellipse doesn't have "magic slopes" -- <br />
you can convince yourself by drawing pictures that, for any slope m, you <br />
can find a line of slope m striking the ellipse twice.<br />
<br />
<br />
[REMARK: For technical reasons we're going to ignore hyperbolae with a <br />
vertical asymptote, since vertical lines don't exactly have slope.]<br />
<br />
<br />
Now you might know the criterion that, over the real numbers.<br />
<br />
<br />
ax^2 + bxy + cy^2 = d<br />
<br />
<br />
is a hyperbola if b^2 - 4ac > 0 and an ellipse if b^2 - 4ac < 0. (For <br />
instance, x^2 - y^2 = 1 is a hyperbola with magic slopes 1 and -1, and x^2 <br />
+ y^2 = 1 is an ellipse.) <br />
This criterion does not make sense in Z/pZ, where there is no notion of <br />
"greater" or "less."<br />
<br />
<br />
QUESTION A.1.<br />
<br />
For which primes p is it the case that <br />
<br />
x^2 + y^2 = 1<br />
<br />
has magic slopes in Z/pZ? <br />
<br />
QUESTION A.2. For which primes p is it the case that<br />
<br />
<br />
x^2 - y^2 = 1<br />
<br />
<br />
has magic slopes in Z/pZ?<br />
<br />
<br />
QUESTION B: Back in the very first week of this course we studied the set of integers x such that x^2 + 1 is prime. Based on the heuristics we discussed in class, about how many x between 1 and N would you expect to satisfy "x^2 + 1 is prime?"<br />
<br />
<br />
QUESTION C: Let z be a complex number. Then the set of all complex numbers of the form a + bz, with a and b integers, forms a lattice in the complex plane (thought of as R^2.) What is the covolume of this lattice, in terms of z?<br />
<br />
<br />
QUESTION D: Let z be a complex number. Using Minkowski's theorem and the result of question C, show that there exist integers a,b, not both zero, such that Norm(a+bz) is at most (4/pi) Im(z). (Hint: let Omega be the set of all complex numbers of norm at most (4/pi) Im(z).)<br />
<br />
<br />
OPTIONAL EXTRA: Can you give an exact formula for the minimal norm of any nonzero complex number of the form a+bz?<br />
<br />
'''December 12''' Book problems 6.1, 6.2, 6.5, 6.10. For extra credit attempt to do problem 6.9.<br />
<br />
--></div>Andreichttps://www.math.wisc.edu/wiki/index.php?title=Math_567_--_Elementary_Number_Theory&diff=19285Math 567 -- Elementary Number Theory2020-03-22T19:05:07Z<p>Andreic: </p>
<hr />
<div>'''MATH 567<br />
<br />
Elementary Number Theory'''<br />
<br />
MWF 1:20-2:10, Van Vleck B119<br />
<br />
'''Professor:''' [http://www.math.wisc.edu/~andreic/ Andrei Caldararu] (andreic@math.wisc.edu)<br />
<br />
''Office Hours:'' Wednesdays 2:30-3:30, Van Vleck 605. <br />
<br />
'''Grader:''' Yifan Peng, email: peng64@wisc.edu<br />
<br />
''Office Hours:'' TBA.<br />
<br />
Math 567 is a course in elementary number theory, aimed at undergraduates majoring in math or other quantitative disciplines. A general familiarity with abstract algebra at the level of Math 541 will be assumed, but students who haven't taken 541 are welcome to attend if they're willing to play a little catchup. We will be using William Stein's new (and cheap) textbook [http://www.amazon.com/Elementary-Number-Theory-Computational-Undergraduate/dp/0387855246 Elementary Number Theory: Primes, Congruences, and Secrets], which emphasizes computational approaches to the subject. If you don't need a physical copy of the book, [http://www.williamstein.org/ent/ it is available as a free legal .pdf.] We will be using the (free, public-domain) mathematical software [http://www.sagemath.org/ SAGE], developed largely by Stein, as an integral component of our coursework. There is a [http://doc.sagemath.org/pdf/en/tutorial/SageTutorial.pdf useful online tutorial.] You can download SAGE to your own computer or [http://www.sagenb.org use it online].<br />
<br />
Topics include some subset of, but are not limited to: Divisibility, the Euclidean algorithm and the GCD, linear Diophantine equations, prime numbers and uniqueness of factorization. Congruences, Chinese remainder theorem, Fermat's "little" theorem, Wilson's theorem, Euler's theorem and totient function, the RSA cryptosystem, Rabin's encryption scheme, Diffie-Hellman key exchange protocol. Number-theoretic functions, multiplicative functions, Möbius inversion. Primitive roots and indices. Quadratic reciprocity and the Legendre symbol. Perfect numbers, Mersenne primes, Fermat primes. Pythagorean triples, Special cases of Fermat's "last" theorem. Fibonacci numbers. Continued fractions. Distribution of primes, discussion of prime number theorem. Primality testing and factoring algorithms.<br />
<br />
'''Course Policies:''' Homework will be due on Fridays. It can be turned in late only with ''advance'' permission from your grader. It is acceptable to use calculators and computers on homework (indeed, some of it will require a computer) but calculators are not allowed during exams. You are encouraged to work together on homework, but writeups must be done individually.<br />
<br />
Many of the problems in this course will ask you to prove things. I expect proofs to be written in English sentences; the proofs in Stein's book are a good model for the level of verbosity I am looking for. <br />
<br />
'''Grading:''' The grade in Math 567 will be composed of 50% homework, 25% midterm, 25% final. The midterm will be on March 11, in class. The final exam will be on 5/8/2020, 7:45AM-9:45AM in room TBA.<br />
<br />
'''Syllabus:''' <br />
(This may change as we see what pace works well for the course. All section numbers refer to Stein's book.)<br />
<br />
* Jan 22-31: Prime numbers, prime factorizations, Euclidean algorithm and GCD (1.1-1.2)<br />
* Feb 3-7: The integers mod n, Euler's theorem, the phi function (2.1-2.2)<br />
* Feb 10-14: Modular exponentiation, primality testing, and primitive roots (2.4-2.5)<br />
* Feb 17-21: Public-key cryptography and RSA (3.1-3.4)<br />
* Feb 24-28: Rabin's algorithm (not in the book); algebraic numbers <br />
* Mar 2-6: Quadratic reciprocity (4.1-4.4)<br />
* Mar 9, Mar 13: Finite and infinite continued fractions (5.1-5.3)<br />
* Mar 11: ''Midterm exam''<br />
* Mar 23-30: Continued fractions and diophantine approximation (5.4-5.5)<br />
* Apr 1-3: Diophantine equations I: Pell's equation and Lagrange's theorem<br />
* Apr 6-10: More diophantine equations, elliptic curves (6.1)<br />
* Apr 13-17: Applications of elliptic curves (6.2-6.3)<br />
* Apr 20-24: More applications of elliptic curves (6.4)<br />
* Apr 27-May 1: advanced topic TBD: maybe additional discussion of cryptographic techniques?<br />
<br />
'''Video Lectures:'''<br />
Due to the university being transitioned to online teaching only, out lectures after spring break will need to be delivered online. I will post here videos for you to watch for the material we need to cover. Please watch them, and please give me feedback on how they could be improved.<br />
<br />
* '''Lecture for Mar. 23''': watch it [https://mediaspace.wisc.edu/media/Math+567+March+23+Lecture/0_fkhvkxxv here].<br />
<br />
'''Homework:'''<br />
Homework is due at the beginning of class on the specified Friday. Typing your homework is not a requirement, but if you don't already know LaTeX I highly recommend that you learn it and use it to typeset your homework. I will sometimes assign extra problems, which I will e-mail to the class list and include here.<br />
<br />
* '''Jan 31''': 1.1, 1.3, 1.5, 1.7 (use SAGE), 1.8, 1.14.<br />
Problem A: Use SAGE to compute the number of x in [1..N] such that x^2 + 1 is prime, for N = 100, N = 1000, and N = 10000. Let f(N) be the number of such x. <br />
<br />
a) Can you formulate a conjecture about the relationship between f(N) and N/log N? <br />
<br />
b) What if x^2 + 1 is replaced with x^2 + 2? Can you explain why x^2 + 2 appears less likely to be prime? (Hint: consider x mod 3.)<br />
<br />
c) Prove that f(N) is at most (1/2)N+1. (Hint: consider x mod 2.) <br />
<br />
d) Give as good an upper bound as you can for f(N).<br />
<br />
Note that, despite the evident regularities you'll observe in this problem, we do not even know whether there are infinitely many primes of the form x^2 + 1! You would become very famous if you proved this.<br />
<br />
* '''Feb 7''': 2.6 (the formulation of numerical evidence should be done by Sage if you've got Sage working, and by calculator if not; you can use an online tool like [http://primes.utm.edu/curios/includes/primetest.php this] to test whether a number is prime.) 2.8,2.9,2.11,2.12,2.14,2.19<br />
<br />
* '''Feb 17''': 2.15, 2.18, 2.23, 2.26, 2.30.<br />
<br />
Problem A: Prove that if n=pq, with p,q prime, then n is not a Carmichael number.<br />
<br />
* '''Feb 21''': Book problems: 3.4, 3.5, 3.6<br />
<br />
Problem A. Prove that there are infinitely many primes p such that 2 is '''not''' a primitive root in Z/pZ. We break this up into steps.<br />
<br />
Problem A.1. Prove that, if x is an element of Z/nZ, then x^2 is not a primitive root.<br />
<br />
Problem A.2. Prove that there are infinitely many primes p such that 2 is a square in Z/pZ. Hint: suppose there are only finitely many such primes p_1, .. p_r, and define N = (p_1 .. p_r)^2 - 2. Where can you go from here...?<br />
<br />
Problem A.3. Give a list of five primes p such that 2 is not a primitive root in Z/pZ (you can use the method of this proof or any other.)<br />
<br />
Problem B. Prove that 24 is the largest integer n such that every element of (Z/nZ)^* is a root of x^2-1.<br />
<br />
* '''Feb 28''': Problem A. Using p = 23 and q=31, show how to encrypt the message x=240 with Rabin's algorithm. Find all possible decryptions of your encrypted message.<br />
<br />
* '''March 23''': 4.1 from the book.<br />
<br />
Problem A. Give a prime factorization of the Gaussian integer 7+9i.<br />
<br />
Problem B. Read the notes from [http://www.math.ucsd.edu/~jverstra/Gaussian1.pdf here]. Note that Z[i] satisfies a ''reduction theorem'': if n and d are Gaussian integers, then there exists integers q and r such that n = qd + r and Norm(r) < Norm(d). But (by contrast with the case of Z) this d may not be unique. In some contexts it is better to be able to choose r uniquely, even if this means letting r have norm greater than Norm(d). <br />
<br />
B.1. When d = 1+2i, show that, for each n in Z[i], there is a '''unique''' pair (q,r) in Z[i] such that n = qd+r and r is contained in the set {0,1,2,3,4}. For instance, i can be written as i(1+2i) + 2, so we say i reduces to 2 mod (1+2i).<br />
(Hint: Suppose q(1+2i)+r = q'(1+2i) + r'. What can we say about (r-r'), and why is this incompatible with both r and r' being in {0,1,2,3,4}?<br />
<br />
B.2. Show that if n is an integer in Z, the reduction of n mod (1+2i) is equal to its reduction mod 5.<br />
<br />
Problem C. Let's try to figure out how to define "phi(d)" for a Gaussian integer d. Suppose S is a set of Gaussian integers such that every n in Z[i] can be written uniquely as qd+r, with q in Z[i] and r in S. (So for instance when d=1+2i, we showed in problem B that we can take S to be {0...4}. It would also be OK to take S to be {1..5} or {0,i,2i,1+i,1+2i}. In fact, it turns out that S has to be a set of size Norm(d) (I might or might not prove this in class; if not, feel free just to accept it.)<br />
<br />
Now define phi(d) to be the number of elements s of S such that s and d are coprime.<br />
<br />
C.1. Compute phi(1+2i) and phi(3).<br />
<br />
C.2. Prove that the value of phi(d) does not depend on the choice of S.<br />
<br />
C.3. Prove that every n in Z[i] which is coprime to 3 satisfies n^phi(3) = 1 mod 3; that is, Euler's theorem holds in this case. (You can prove this by direct computation; of course, if you want, you are welcome to prove that Euler's theorem holds for Z[i] in general, adapting the proof in Stein's book or the one we gave in class.)<br />
<br />
<!--<br />
*'''Nov 2:''' <br />
<br />
Problem A. Express 50005 as the sum of two squares.<br />
<br />
In the next two problems we denote by r(n) the number of ways to express n as the sum of two squares (i.e. the number of pairs (a,b) such that a^2 + b^2 = n.) For instance, r(5) = 8 (as shown on the midterm.)<br />
<br />
Problem B. Prove that, for any N, there exists an integer n such that r(n) > N. (I.E., the function r(n) is ''unbounded.''<br />
<br />
Problem C. If you like Sage, write a short program in Sage to compute r(n) and compute the average of r(n) as n ranges from 1 to 1000. Whether or not you like Sage, make a guess as to how this average would behave if you replaced "1000" by a larger and larger number. (Feel free to ask the Sage-lovers what answer they got in the optional first part of the question.) Can you prove this guess is correct?<br />
<br />
Problem D. We saw in class that the ring Z[sqrt(-5)] doesn't have unique factorization; 6 can be factored as 2*3 or (1+sqrt(-5))(1-sqrt(-5)). In this problem, we will prove that Z[sqrt(-d)] fails to have unique factorization for EVERY odd d >= 5. (Actually it's true for all d >= 5 but to make the proof manageable we'll restrict to the odd case.]<br />
<br />
D.1. Show that (1+sqrt(-d)), (1-sqrt(-d)) and 2 are irreducible in Z[sqrt(-d)].<br />
<br />
D.2. Now give an element of Z[sqrt(-d)] that has two distinct factorizations into irreducibles. (Hint: imitate the example we used for Z[sqrt(-5)].)<br />
''Remark:'' The rings Z[sqrt(d)], where d is positive, are quite different -- here we believe that there are infinitely many with unique factorization, though this conjecture has remained unproved for many decades!<br />
<br />
*'''November 16'''<br />
<br />
Book problems: 5.3,5.4,5.5<br />
<br />
Problem A: We discussed in class the problem of studying which positive integers are the sum of two squares. In this problem we prove that every element n of (Z/pZ) is the sum of two squares. We argue as follows. Let S be the set of squares in (Z/pZ), and let T be the set of (Z/pZ) consisting of all elements of the form n - x^2, for some x in (Z/pZ).<br />
<br />
A.1. What is the size of S and of T? Use this to show that S and T are not disjoint.<br />
<br />
A.2. Given that S and T are not disjoint, prove that n is the sum of two squares.<br />
<br />
Problem B. Using the continued fraction expansion, find a solution to the Pell equation x^2 - 13 y^2 = 1.<br />
<br />
Problem C. Show that the modified Pell equation x^2 - 7y^2 = -1 has no solutions in integers x,y. (Hint: reduce the equation modulo a suitably chosen prime.)<br />
<br />
Problem D. Stuffy Stirnweiss finished the 1945 season with a batting average of .3085443. Using continued fractions, guess how many at-bats he had. Tony Cuccinello had a batting average of .3084577. Given that he had more than 200 and fewer than 600 at-bats, can you estimate the number of at-bats he had?<br />
<br />
'''December 7'''<br />
<br />
Problem A. When p is a prime congruent to 3 mod 4, prove that ((p-1)/2)! is either 1 or -1 in (Z/pZ). '''OPTIONAL:''' Use sage to compute whether this factorial is 1 or -1 for many primes p. Is there any pattern? Does it seem to be 1 half the time and -1 half the time? Note that I have no idea what the answer to this question is.<br />
<br />
Problem B. Using the continued fraction expansion, find a solution to the Pell equation x^2 - 13 y^2 = 1.<br />
<br />
Problem C. Show that the modified Pell equation x^2 - 7y^2 = -1 has no solutions in integers x,y. (Hint: reduce the equation modulo a suitably chosen prime.)<br />
<br />
Problem D.1. Pick two values of a in F_11 = Z/11Z (a not equal to 3), such that the equation y^2 = x^3+ax+1 defines an elliptic curve (i.e., it is smooth). For each such a, determine the number of points #E(F_11) and check that it falls inside the interval described in class. <br />
<br />
Problem D.2. The point P = (0,1) lies on each of these curves. For a =3 (the curve discussed in class) determine the order of P in the elliptic curve group, that is, find the smallest positive integer n such that nP = (infinity) -- the identity element in the group law.<br />
<br />
<!-- '''November 19'''<br />
<br />
Problem A. Using the method discussed in class (which is also the method of problem 4.6 in Stein, which in retrospect I think was too hard to assign with no preparation) find a nontrivial solution with y > 0 to the equation x^2 + 11y^2 = z^2. <br />
Problem B. Let f(X) be the number of solutions of A^2 + B^2 = C^3 such that A^2, B^2, and C^3 are all at most X. Use the heuristic described in class to explain why one might expect f(X) to grow more or less like X^{1/3}.<br />
<br />
Problem C. Show that if A+Bi is the cube of a Gaussian integer, then A^2 + B^2 is a perfect cube.<br />
<br />
Do ONE of problem D1 and D2; D1 is for people who like Sage, D2 is for people who like proving things.<br />
<br />
D1. Use Sage to compute f(X) for X = 1000, 10000, 100000. Does the answer look consistent with the heuristic prediction you made in B? Does f(X)/X^{1/3} appear to be approaching a limit?<br />
D2. We will use problem C to give a lower bound for f(X). Use this to show that there are at least X^{1/3} Gaussian integers with norm at most X that are perfect cubes. From here, show that f(X) > X^{1/3}.<br />
<br />
OPTIONAL: Show that the converse of problem C also holds -- A+Bi has norm a perfect cube if and only if A+Bi is a perfect cube. (You will need unique factorization of Gaussian integers.) Using this fact, prove that f(X) is bounded above by C X^{1/3} for some constant C.<br />
<br />
'''December 8 (note nonstandard due date)'''<br />
<br />
Problem A: Hyperbolas, ellipses, and "magic slopes"<br />
<br />
In the usual analytic geometry, both hyperbolas and ellipses are given by equations of the form<br />
<br />
ax^2 + bxy + cy^2 = d (*)<br />
<br />
<br />
(we always assume d is nonzero.)<br />
<br />
How can we tell whether such an equation describes an ellipse or a<br />
hyperbola? Well, in the geometric setting we all know and love (i.e. in<br />
R^2) a hyperbola has asymptotes and an ellipse does not. What is an <br />
asymptote? You could say it's "a line which doesn't intersect the curve", <br />
but ellipses have such lines too. You might want to say "a line which <br />
doesn't intersect the curve but comes closer and closer to the curve," but <br />
the problem here is that this a) is somewhat imprecise, and b) relies on a <br />
notion of "closer" that is not going to be very clear in Z/pZ, which is <br />
our ultimate goal!<br />
<br />
<br />
So let me state it a different way: let's say the asymptotes of a <br />
hyperbola have slopes m_1 and m_2. These are "magic slopes" in the <br />
following sense: ANY line of slope m_1 (and ditto for m_2) intersects the <br />
hyperbola in at most one point. An ellipse doesn't have "magic slopes" -- <br />
you can convince yourself by drawing pictures that, for any slope m, you <br />
can find a line of slope m striking the ellipse twice.<br />
<br />
<br />
[REMARK: For technical reasons we're going to ignore hyperbolae with a <br />
vertical asymptote, since vertical lines don't exactly have slope.]<br />
<br />
<br />
Now you might know the criterion that, over the real numbers.<br />
<br />
<br />
ax^2 + bxy + cy^2 = d<br />
<br />
<br />
is a hyperbola if b^2 - 4ac > 0 and an ellipse if b^2 - 4ac < 0. (For <br />
instance, x^2 - y^2 = 1 is a hyperbola with magic slopes 1 and -1, and x^2 <br />
+ y^2 = 1 is an ellipse.) <br />
This criterion does not make sense in Z/pZ, where there is no notion of <br />
"greater" or "less."<br />
<br />
<br />
QUESTION A.1.<br />
<br />
For which primes p is it the case that <br />
<br />
x^2 + y^2 = 1<br />
<br />
has magic slopes in Z/pZ? <br />
<br />
QUESTION A.2. For which primes p is it the case that<br />
<br />
<br />
x^2 - y^2 = 1<br />
<br />
<br />
has magic slopes in Z/pZ?<br />
<br />
<br />
QUESTION B: Back in the very first week of this course we studied the set of integers x such that x^2 + 1 is prime. Based on the heuristics we discussed in class, about how many x between 1 and N would you expect to satisfy "x^2 + 1 is prime?"<br />
<br />
<br />
QUESTION C: Let z be a complex number. Then the set of all complex numbers of the form a + bz, with a and b integers, forms a lattice in the complex plane (thought of as R^2.) What is the covolume of this lattice, in terms of z?<br />
<br />
<br />
QUESTION D: Let z be a complex number. Using Minkowski's theorem and the result of question C, show that there exist integers a,b, not both zero, such that Norm(a+bz) is at most (4/pi) Im(z). (Hint: let Omega be the set of all complex numbers of norm at most (4/pi) Im(z).)<br />
<br />
<br />
OPTIONAL EXTRA: Can you give an exact formula for the minimal norm of any nonzero complex number of the form a+bz?<br />
<br />
'''December 12''' Book problems 6.1, 6.2, 6.5, 6.10. For extra credit attempt to do problem 6.9.<br />
<br />
--></div>Andreichttps://www.math.wisc.edu/wiki/index.php?title=Math_567_--_Elementary_Number_Theory&diff=19225Math 567 -- Elementary Number Theory2020-03-09T17:44:45Z<p>Andreic: </p>
<hr />
<div>'''MATH 567<br />
<br />
Elementary Number Theory'''<br />
<br />
MWF 1:20-2:10, Van Vleck B119<br />
<br />
'''Professor:''' [http://www.math.wisc.edu/~andreic/ Andrei Caldararu] (andreic@math.wisc.edu)<br />
<br />
''Office Hours:'' Wednesdays 2:30-3:30, Van Vleck 605. <br />
<br />
'''Grader:''' Yifan Peng, email: peng64@wisc.edu<br />
<br />
''Office Hours:'' TBA.<br />
<br />
Math 567 is a course in elementary number theory, aimed at undergraduates majoring in math or other quantitative disciplines. A general familiarity with abstract algebra at the level of Math 541 will be assumed, but students who haven't taken 541 are welcome to attend if they're willing to play a little catchup. We will be using William Stein's new (and cheap) textbook [http://www.amazon.com/Elementary-Number-Theory-Computational-Undergraduate/dp/0387855246 Elementary Number Theory: Primes, Congruences, and Secrets], which emphasizes computational approaches to the subject. If you don't need a physical copy of the book, [http://www.williamstein.org/ent/ it is available as a free legal .pdf.] We will be using the (free, public-domain) mathematical software [http://www.sagemath.org/ SAGE], developed largely by Stein, as an integral component of our coursework. There is a [http://doc.sagemath.org/pdf/en/tutorial/SageTutorial.pdf useful online tutorial.] You can download SAGE to your own computer or [http://www.sagenb.org use it online].<br />
<br />
Topics include some subset of, but are not limited to: Divisibility, the Euclidean algorithm and the GCD, linear Diophantine equations, prime numbers and uniqueness of factorization. Congruences, Chinese remainder theorem, Fermat's "little" theorem, Wilson's theorem, Euler's theorem and totient function, the RSA cryptosystem, Rabin's encryption scheme, Diffie-Hellman key exchange protocol. Number-theoretic functions, multiplicative functions, Möbius inversion. Primitive roots and indices. Quadratic reciprocity and the Legendre symbol. Perfect numbers, Mersenne primes, Fermat primes. Pythagorean triples, Special cases of Fermat's "last" theorem. Fibonacci numbers. Continued fractions. Distribution of primes, discussion of prime number theorem. Primality testing and factoring algorithms.<br />
<br />
'''Course Policies:''' Homework will be due on Fridays. It can be turned in late only with ''advance'' permission from your grader. It is acceptable to use calculators and computers on homework (indeed, some of it will require a computer) but calculators are not allowed during exams. You are encouraged to work together on homework, but writeups must be done individually.<br />
<br />
Many of the problems in this course will ask you to prove things. I expect proofs to be written in English sentences; the proofs in Stein's book are a good model for the level of verbosity I am looking for. <br />
<br />
'''Grading:''' The grade in Math 567 will be composed of 50% homework, 25% midterm, 25% final. The midterm will be on March 11, in class. The final exam will be on 5/8/2020, 7:45AM-9:45AM in room TBA.<br />
<br />
'''Syllabus:''' <br />
(This may change as we see what pace works well for the course. All section numbers refer to Stein's book.)<br />
<br />
* Jan 22-31: Prime numbers, prime factorizations, Euclidean algorithm and GCD (1.1-1.2)<br />
* Feb 3-7: The integers mod n, Euler's theorem, the phi function (2.1-2.2)<br />
* Feb 10-14: Modular exponentiation, primality testing, and primitive roots (2.4-2.5)<br />
* Feb 17-21: Public-key cryptography and RSA (3.1-3.4)<br />
* Feb 24-28: Rabin's algorithm (not in the book); algebraic numbers <br />
* Mar 2-6: Quadratic reciprocity (4.1-4.4)<br />
* Mar 9, Mar 13: Finite and infinite continued fractions (5.1-5.3)<br />
* Mar 11: ''Midterm exam''<br />
* Mar 23-30: Continued fractions and diophantine approximation (5.4-5.5)<br />
* Apr 1-3: Diophantine equations I: Pell's equation and Lagrange's theorem<br />
* Apr 6-10: More diophantine equations, elliptic curves (6.1)<br />
* Apr 13-17: Applications of elliptic curves (6.2-6.3)<br />
* Apr 20-24: More applications of elliptic curves (6.4)<br />
* Apr 27-May 1: advanced topic TBD: maybe additional discussion of cryptographic techniques?<br />
<br />
'''Homework:'''<br />
Homework is due at the beginning of class on the specified Friday. Typing your homework is not a requirement, but if you don't already know LaTeX I highly recommend that you learn it and use it to typeset your homework. I will sometimes assign extra problems, which I will e-mail to the class list and include here.<br />
<br />
* '''Jan 31''': 1.1, 1.3, 1.5, 1.7 (use SAGE), 1.8, 1.14.<br />
Problem A: Use SAGE to compute the number of x in [1..N] such that x^2 + 1 is prime, for N = 100, N = 1000, and N = 10000. Let f(N) be the number of such x. <br />
<br />
a) Can you formulate a conjecture about the relationship between f(N) and N/log N? <br />
<br />
b) What if x^2 + 1 is replaced with x^2 + 2? Can you explain why x^2 + 2 appears less likely to be prime? (Hint: consider x mod 3.)<br />
<br />
c) Prove that f(N) is at most (1/2)N+1. (Hint: consider x mod 2.) <br />
<br />
d) Give as good an upper bound as you can for f(N).<br />
<br />
Note that, despite the evident regularities you'll observe in this problem, we do not even know whether there are infinitely many primes of the form x^2 + 1! You would become very famous if you proved this.<br />
<br />
* '''Feb 7''': 2.6 (the formulation of numerical evidence should be done by Sage if you've got Sage working, and by calculator if not; you can use an online tool like [http://primes.utm.edu/curios/includes/primetest.php this] to test whether a number is prime.) 2.8,2.9,2.11,2.12,2.14,2.19<br />
<br />
* '''Feb 17''': 2.15, 2.18, 2.23, 2.26, 2.30.<br />
<br />
Problem A: Prove that if n=pq, with p,q prime, then n is not a Carmichael number.<br />
<br />
* '''Feb 21''': Book problems: 3.4, 3.5, 3.6<br />
<br />
Problem A. Prove that there are infinitely many primes p such that 2 is '''not''' a primitive root in Z/pZ. We break this up into steps.<br />
<br />
Problem A.1. Prove that, if x is an element of Z/nZ, then x^2 is not a primitive root.<br />
<br />
Problem A.2. Prove that there are infinitely many primes p such that 2 is a square in Z/pZ. Hint: suppose there are only finitely many such primes p_1, .. p_r, and define N = (p_1 .. p_r)^2 - 2. Where can you go from here...?<br />
<br />
Problem A.3. Give a list of five primes p such that 2 is not a primitive root in Z/pZ (you can use the method of this proof or any other.)<br />
<br />
Problem B. Prove that 24 is the largest integer n such that every element of (Z/nZ)^* is a root of x^2-1.<br />
<br />
* '''Feb 28''': Problem A. Using p = 23 and q=31, show how to encrypt the message x=240 with Rabin's algorithm. Find all possible decryptions of your encrypted message.<br />
<br />
* '''March 23''': 4.1 from the book.<br />
<br />
Problem A. Give a prime factorization of the Gaussian integer 7+9i.<br />
<br />
Problem B. Read the notes from [http://www.math.ucsd.edu/~jverstra/Gaussian1.pdf here]. Note that Z[i] satisfies a ''reduction theorem'': if n and d are Gaussian integers, then there exists integers q and r such that n = qd + r and Norm(r) < Norm(d). But (by contrast with the case of Z) this d may not be unique. In some contexts it is better to be able to choose r uniquely, even if this means letting r have norm greater than Norm(d). <br />
<br />
B.1. When d = 1+2i, show that, for each n in Z[i], there is a '''unique''' pair (q,r) in Z[i] such that n = qd+r and r is contained in the set {0,1,2,3,4}. For instance, i can be written as i(1+2i) + 2, so we say i reduces to 2 mod (1+2i).<br />
(Hint: Suppose q(1+2i)+r = q'(1+2i) + r'. What can we say about (r-r'), and why is this incompatible with both r and r' being in {0,1,2,3,4}?<br />
<br />
B.2. Show that if n is an integer in Z, the reduction of n mod (1+2i) is equal to its reduction mod 5.<br />
<br />
Problem C. Let's try to figure out how to define "phi(d)" for a Gaussian integer d. Suppose S is a set of Gaussian integers such that every n in Z[i] can be written uniquely as qd+r, with q in Z[i] and r in S. (So for instance when d=1+2i, we showed in problem B that we can take S to be {0...4}. It would also be OK to take S to be {1..5} or {0,i,2i,1+i,1+2i}. In fact, it turns out that S has to be a set of size Norm(d) (I might or might not prove this in class; if not, feel free just to accept it.)<br />
<br />
Now define phi(d) to be the number of elements s of S such that s and d are coprime.<br />
<br />
C.1. Compute phi(1+2i) and phi(3).<br />
<br />
C.2. Prove that the value of phi(d) does not depend on the choice of S.<br />
<br />
C.3. Prove that every n in Z[i] which is coprime to 3 satisfies n^phi(3) = 1 mod 3; that is, Euler's theorem holds in this case. (You can prove this by direct computation; of course, if you want, you are welcome to prove that Euler's theorem holds for Z[i] in general, adapting the proof in Stein's book or the one we gave in class.)<br />
<br />
<!--<br />
*'''Nov 2:''' <br />
<br />
Problem A. Express 50005 as the sum of two squares.<br />
<br />
In the next two problems we denote by r(n) the number of ways to express n as the sum of two squares (i.e. the number of pairs (a,b) such that a^2 + b^2 = n.) For instance, r(5) = 8 (as shown on the midterm.)<br />
<br />
Problem B. Prove that, for any N, there exists an integer n such that r(n) > N. (I.E., the function r(n) is ''unbounded.''<br />
<br />
Problem C. If you like Sage, write a short program in Sage to compute r(n) and compute the average of r(n) as n ranges from 1 to 1000. Whether or not you like Sage, make a guess as to how this average would behave if you replaced "1000" by a larger and larger number. (Feel free to ask the Sage-lovers what answer they got in the optional first part of the question.) Can you prove this guess is correct?<br />
<br />
Problem D. We saw in class that the ring Z[sqrt(-5)] doesn't have unique factorization; 6 can be factored as 2*3 or (1+sqrt(-5))(1-sqrt(-5)). In this problem, we will prove that Z[sqrt(-d)] fails to have unique factorization for EVERY odd d >= 5. (Actually it's true for all d >= 5 but to make the proof manageable we'll restrict to the odd case.]<br />
<br />
D.1. Show that (1+sqrt(-d)), (1-sqrt(-d)) and 2 are irreducible in Z[sqrt(-d)].<br />
<br />
D.2. Now give an element of Z[sqrt(-d)] that has two distinct factorizations into irreducibles. (Hint: imitate the example we used for Z[sqrt(-5)].)<br />
''Remark:'' The rings Z[sqrt(d)], where d is positive, are quite different -- here we believe that there are infinitely many with unique factorization, though this conjecture has remained unproved for many decades!<br />
<br />
*'''November 16'''<br />
<br />
Book problems: 5.3,5.4,5.5<br />
<br />
Problem A: We discussed in class the problem of studying which positive integers are the sum of two squares. In this problem we prove that every element n of (Z/pZ) is the sum of two squares. We argue as follows. Let S be the set of squares in (Z/pZ), and let T be the set of (Z/pZ) consisting of all elements of the form n - x^2, for some x in (Z/pZ).<br />
<br />
A.1. What is the size of S and of T? Use this to show that S and T are not disjoint.<br />
<br />
A.2. Given that S and T are not disjoint, prove that n is the sum of two squares.<br />
<br />
Problem B. Using the continued fraction expansion, find a solution to the Pell equation x^2 - 13 y^2 = 1.<br />
<br />
Problem C. Show that the modified Pell equation x^2 - 7y^2 = -1 has no solutions in integers x,y. (Hint: reduce the equation modulo a suitably chosen prime.)<br />
<br />
Problem D. Stuffy Stirnweiss finished the 1945 season with a batting average of .3085443. Using continued fractions, guess how many at-bats he had. Tony Cuccinello had a batting average of .3084577. Given that he had more than 200 and fewer than 600 at-bats, can you estimate the number of at-bats he had?<br />
<br />
'''December 7'''<br />
<br />
Problem A. When p is a prime congruent to 3 mod 4, prove that ((p-1)/2)! is either 1 or -1 in (Z/pZ). '''OPTIONAL:''' Use sage to compute whether this factorial is 1 or -1 for many primes p. Is there any pattern? Does it seem to be 1 half the time and -1 half the time? Note that I have no idea what the answer to this question is.<br />
<br />
Problem B. Using the continued fraction expansion, find a solution to the Pell equation x^2 - 13 y^2 = 1.<br />
<br />
Problem C. Show that the modified Pell equation x^2 - 7y^2 = -1 has no solutions in integers x,y. (Hint: reduce the equation modulo a suitably chosen prime.)<br />
<br />
Problem D.1. Pick two values of a in F_11 = Z/11Z (a not equal to 3), such that the equation y^2 = x^3+ax+1 defines an elliptic curve (i.e., it is smooth). For each such a, determine the number of points #E(F_11) and check that it falls inside the interval described in class. <br />
<br />
Problem D.2. The point P = (0,1) lies on each of these curves. For a =3 (the curve discussed in class) determine the order of P in the elliptic curve group, that is, find the smallest positive integer n such that nP = (infinity) -- the identity element in the group law.<br />
<br />
<!-- '''November 19'''<br />
<br />
Problem A. Using the method discussed in class (which is also the method of problem 4.6 in Stein, which in retrospect I think was too hard to assign with no preparation) find a nontrivial solution with y > 0 to the equation x^2 + 11y^2 = z^2. <br />
Problem B. Let f(X) be the number of solutions of A^2 + B^2 = C^3 such that A^2, B^2, and C^3 are all at most X. Use the heuristic described in class to explain why one might expect f(X) to grow more or less like X^{1/3}.<br />
<br />
Problem C. Show that if A+Bi is the cube of a Gaussian integer, then A^2 + B^2 is a perfect cube.<br />
<br />
Do ONE of problem D1 and D2; D1 is for people who like Sage, D2 is for people who like proving things.<br />
<br />
D1. Use Sage to compute f(X) for X = 1000, 10000, 100000. Does the answer look consistent with the heuristic prediction you made in B? Does f(X)/X^{1/3} appear to be approaching a limit?<br />
D2. We will use problem C to give a lower bound for f(X). Use this to show that there are at least X^{1/3} Gaussian integers with norm at most X that are perfect cubes. From here, show that f(X) > X^{1/3}.<br />
<br />
OPTIONAL: Show that the converse of problem C also holds -- A+Bi has norm a perfect cube if and only if A+Bi is a perfect cube. (You will need unique factorization of Gaussian integers.) Using this fact, prove that f(X) is bounded above by C X^{1/3} for some constant C.<br />
<br />
'''December 8 (note nonstandard due date)'''<br />
<br />
Problem A: Hyperbolas, ellipses, and "magic slopes"<br />
<br />
In the usual analytic geometry, both hyperbolas and ellipses are given by equations of the form<br />
<br />
ax^2 + bxy + cy^2 = d (*)<br />
<br />
<br />
(we always assume d is nonzero.)<br />
<br />
How can we tell whether such an equation describes an ellipse or a<br />
hyperbola? Well, in the geometric setting we all know and love (i.e. in<br />
R^2) a hyperbola has asymptotes and an ellipse does not. What is an <br />
asymptote? You could say it's "a line which doesn't intersect the curve", <br />
but ellipses have such lines too. You might want to say "a line which <br />
doesn't intersect the curve but comes closer and closer to the curve," but <br />
the problem here is that this a) is somewhat imprecise, and b) relies on a <br />
notion of "closer" that is not going to be very clear in Z/pZ, which is <br />
our ultimate goal!<br />
<br />
<br />
So let me state it a different way: let's say the asymptotes of a <br />
hyperbola have slopes m_1 and m_2. These are "magic slopes" in the <br />
following sense: ANY line of slope m_1 (and ditto for m_2) intersects the <br />
hyperbola in at most one point. An ellipse doesn't have "magic slopes" -- <br />
you can convince yourself by drawing pictures that, for any slope m, you <br />
can find a line of slope m striking the ellipse twice.<br />
<br />
<br />
[REMARK: For technical reasons we're going to ignore hyperbolae with a <br />
vertical asymptote, since vertical lines don't exactly have slope.]<br />
<br />
<br />
Now you might know the criterion that, over the real numbers.<br />
<br />
<br />
ax^2 + bxy + cy^2 = d<br />
<br />
<br />
is a hyperbola if b^2 - 4ac > 0 and an ellipse if b^2 - 4ac < 0. (For <br />
instance, x^2 - y^2 = 1 is a hyperbola with magic slopes 1 and -1, and x^2 <br />
+ y^2 = 1 is an ellipse.) <br />
This criterion does not make sense in Z/pZ, where there is no notion of <br />
"greater" or "less."<br />
<br />
<br />
QUESTION A.1.<br />
<br />
For which primes p is it the case that <br />
<br />
x^2 + y^2 = 1<br />
<br />
has magic slopes in Z/pZ? <br />
<br />
QUESTION A.2. For which primes p is it the case that<br />
<br />
<br />
x^2 - y^2 = 1<br />
<br />
<br />
has magic slopes in Z/pZ?<br />
<br />
<br />
QUESTION B: Back in the very first week of this course we studied the set of integers x such that x^2 + 1 is prime. Based on the heuristics we discussed in class, about how many x between 1 and N would you expect to satisfy "x^2 + 1 is prime?"<br />
<br />
<br />
QUESTION C: Let z be a complex number. Then the set of all complex numbers of the form a + bz, with a and b integers, forms a lattice in the complex plane (thought of as R^2.) What is the covolume of this lattice, in terms of z?<br />
<br />
<br />
QUESTION D: Let z be a complex number. Using Minkowski's theorem and the result of question C, show that there exist integers a,b, not both zero, such that Norm(a+bz) is at most (4/pi) Im(z). (Hint: let Omega be the set of all complex numbers of norm at most (4/pi) Im(z).)<br />
<br />
<br />
OPTIONAL EXTRA: Can you give an exact formula for the minimal norm of any nonzero complex number of the form a+bz?<br />
<br />
'''December 12''' Book problems 6.1, 6.2, 6.5, 6.10. For extra credit attempt to do problem 6.9.<br />
<br />
--></div>Andreichttps://www.math.wisc.edu/wiki/index.php?title=Math_567_--_Elementary_Number_Theory&diff=19224Math 567 -- Elementary Number Theory2020-03-09T17:40:25Z<p>Andreic: </p>
<hr />
<div>'''MATH 567<br />
<br />
Elementary Number Theory'''<br />
<br />
MWF 1:20-2:10, Van Vleck B119<br />
<br />
'''Professor:''' [http://www.math.wisc.edu/~andreic/ Andrei Caldararu] (andreic@math.wisc.edu)<br />
<br />
''Office Hours:'' Wednesdays 2:30-3:30, Van Vleck 605. <br />
<br />
'''Grader:''' Yifan Peng, email: peng64@wisc.edu<br />
<br />
''Office Hours:'' TBA.<br />
<br />
Math 567 is a course in elementary number theory, aimed at undergraduates majoring in math or other quantitative disciplines. A general familiarity with abstract algebra at the level of Math 541 will be assumed, but students who haven't taken 541 are welcome to attend if they're willing to play a little catchup. We will be using William Stein's new (and cheap) textbook [http://www.amazon.com/Elementary-Number-Theory-Computational-Undergraduate/dp/0387855246 Elementary Number Theory: Primes, Congruences, and Secrets], which emphasizes computational approaches to the subject. If you don't need a physical copy of the book, [http://www.williamstein.org/ent/ it is available as a free legal .pdf.] We will be using the (free, public-domain) mathematical software [http://www.sagemath.org/ SAGE], developed largely by Stein, as an integral component of our coursework. There is a [http://doc.sagemath.org/pdf/en/tutorial/SageTutorial.pdf useful online tutorial.] You can download SAGE to your own computer or [http://www.sagenb.org use it online].<br />
<br />
Topics include some subset of, but are not limited to: Divisibility, the Euclidean algorithm and the GCD, linear Diophantine equations, prime numbers and uniqueness of factorization. Congruences, Chinese remainder theorem, Fermat's "little" theorem, Wilson's theorem, Euler's theorem and totient function, the RSA cryptosystem, Rabin's encryption scheme, Diffie-Hellman key exchange protocol. Number-theoretic functions, multiplicative functions, Möbius inversion. Primitive roots and indices. Quadratic reciprocity and the Legendre symbol. Perfect numbers, Mersenne primes, Fermat primes. Pythagorean triples, Special cases of Fermat's "last" theorem. Fibonacci numbers. Continued fractions. Distribution of primes, discussion of prime number theorem. Primality testing and factoring algorithms.<br />
<br />
'''Course Policies:''' Homework will be due on Fridays. It can be turned in late only with ''advance'' permission from your grader. It is acceptable to use calculators and computers on homework (indeed, some of it will require a computer) but calculators are not allowed during exams. You are encouraged to work together on homework, but writeups must be done individually.<br />
<br />
Many of the problems in this course will ask you to prove things. I expect proofs to be written in English sentences; the proofs in Stein's book are a good model for the level of verbosity I am looking for. <br />
<br />
'''Grading:''' The grade in Math 567 will be composed of 50% homework, 25% midterm, 25% final. The midterm will be on March 11, in class. The final exam will be on 5/8/2020, 7:45AM-9:45AM in room TBA.<br />
<br />
'''Syllabus:''' <br />
(This may change as we see what pace works well for the course. All section numbers refer to Stein's book.)<br />
<br />
* Jan 22-31: Prime numbers, prime factorizations, Euclidean algorithm and GCD (1.1-1.2)<br />
* Feb 3-7: The integers mod n, Euler's theorem, the phi function (2.1-2.2)<br />
* Feb 10-14: Modular exponentiation, primality testing, and primitive roots (2.4-2.5)<br />
* Feb 17-21: Public-key cryptography and RSA (3.1-3.4)<br />
* Feb 24-28: Rabin's algorithm (not in the book); algebraic numbers <br />
* Mar 2-6: Quadratic reciprocity (4.1-4.4)<br />
* Mar 9, Mar 13: Finite and infinite continued fractions (5.1-5.3)<br />
* Mar 11: ''Midterm exam''<br />
* Mar 23-30: Continued fractions and diophantine approximation (5.4-5.5)<br />
* Apr 1-3: Diophantine equations I: Pell's equation and Lagrange's theorem<br />
* Apr 6-10: More diophantine equations, elliptic curves (6.1)<br />
* Apr 13-17: Applications of elliptic curves (6.2-6.3)<br />
* Apr 20-24: More applications of elliptic curves (6.4)<br />
* Apr 27-May 1: advanced topic TBD: maybe additional discussion of cryptographic techniques?<br />
<br />
'''Homework:'''<br />
Homework is due at the beginning of class on the specified Friday. Typing your homework is not a requirement, but if you don't already know LaTeX I highly recommend that you learn it and use it to typeset your homework. I will sometimes assign extra problems, which I will e-mail to the class list and include here.<br />
<br />
* '''Jan 31''': 1.1, 1.3, 1.5, 1.7 (use SAGE), 1.8, 1.14.<br />
Problem A: Use SAGE to compute the number of x in [1..N] such that x^2 + 1 is prime, for N = 100, N = 1000, and N = 10000. Let f(N) be the number of such x. <br />
<br />
a) Can you formulate a conjecture about the relationship between f(N) and N/log N? <br />
<br />
b) What if x^2 + 1 is replaced with x^2 + 2? Can you explain why x^2 + 2 appears less likely to be prime? (Hint: consider x mod 3.)<br />
<br />
c) Prove that f(N) is at most (1/2)N+1. (Hint: consider x mod 2.) <br />
<br />
d) Give as good an upper bound as you can for f(N).<br />
<br />
Note that, despite the evident regularities you'll observe in this problem, we do not even know whether there are infinitely many primes of the form x^2 + 1! You would become very famous if you proved this.<br />
<br />
* '''Feb 7''': 2.6 (the formulation of numerical evidence should be done by Sage if you've got Sage working, and by calculator if not; you can use an online tool like [http://primes.utm.edu/curios/includes/primetest.php this] to test whether a number is prime.) 2.8,2.9,2.11,2.12,2.14,2.19<br />
<br />
* '''Feb 17''': 2.15, 2.18, 2.23, 2.26, 2.30.<br />
<br />
Problem A: Prove that if n=pq, with p,q prime, then n is not a Carmichael number.<br />
<br />
* '''Feb 21''': Book problems: 3.4, 3.5, 3.6<br />
<br />
Problem A. Prove that there are infinitely many primes p such that 2 is '''not''' a primitive root in Z/pZ. We break this up into steps.<br />
<br />
Problem A.1. Prove that, if x is an element of Z/nZ, then x^2 is not a primitive root.<br />
<br />
Problem A.2. Prove that there are infinitely many primes p such that 2 is a square in Z/pZ. Hint: suppose there are only finitely many such primes p_1, .. p_r, and define N = (p_1 .. p_r)^2 - 2. Where can you go from here...?<br />
<br />
Problem A.3. Give a list of five primes p such that 2 is not a primitive root in Z/pZ (you can use the method of this proof or any other.)<br />
<br />
Problem B. Prove that 24 is the largest integer n such that every element of (Z/nZ)^* is a root of x^2-1.<br />
<br />
* '''Feb 28''': Problem A. Using p = 23 and q=31, show how to encrypt the message x=240 with Rabin's algorithm. Find all possible decryptions of your encrypted message.<br />
<br />
* '''March 13''': 4.1 from the book.<br />
<br />
Problem A. Give a prime factorization of the Gaussian integer 7+9i.<br />
<br />
Problem B. Read the notes from [http://www.math.ucsd.edu/~jverstra/Gaussian1.pdf here]. Note that Z[i] satisfies a ''reduction theorem'': if n and d are Gaussian integers, then there exists integers q and r such that n = qd + r and Norm(r) < Norm(d). But (by contrast with the case of Z) this d may not be unique. In some contexts it is better to be able to choose r uniquely, even if this means letting r have norm greater than Norm(d). <br />
<br />
B.1. When d = 1+2i, show that, for each n in Z[i], there is a '''unique''' pair (q,r) in Z[i] such that n = qd+r and r is contained in the set {0,1,2,3,4}. For instance, i can be written as i(1+2i) + 2, so we say i reduces to 2 mod (1+2i).<br />
(Hint: Suppose q(1+2i)+r = q'(1+2i) + r'. What can we say about (r-r'), and why is this incompatible with both r and r' being in {0,1,2,3,4}?<br />
<br />
B.2. Show that if n is an integer in Z, the reduction of n mod (1+2i) is equal to its reduction mod 5.<br />
<br />
Problem C. Let's try to figure out how to define "phi(d)" for a Gaussian integer d. Suppose S is a set of Gaussian integers such that every n in Z[i] can be written uniquely as qd+r, with q in Z[i] and r in S. (So for instance when d=1+2i, we showed in problem B that we can take S to be {0...4}. It would also be OK to take S to be {1..5} or {0,i,2i,1+i,1+2i}. In fact, it turns out that S has to be a set of size Norm(d) (I might or might not prove this in class; if not, feel free just to accept it.)<br />
<br />
Now define phi(d) to be the number of elements s of S such that s and d are coprime.<br />
<br />
C.1. Compute phi(1+2i) and phi(3).<br />
<br />
C.2. Prove that the value of phi(d) does not depend on the choice of S.<br />
<br />
C.3. Prove that every n in Z[i] which is coprime to 3 satisfies n^phi(3) = 1 mod 3; that is, Euler's theorem holds in this case. (You can prove this by direct computation; of course, if you want, you are welcome to prove that Euler's theorem holds for Z[i] in general, adapting the proof in Stein's book or the one we gave in class.)<br />
<br />
<!--<br />
*'''Nov 2:''' <br />
<br />
Problem A. Express 50005 as the sum of two squares.<br />
<br />
In the next two problems we denote by r(n) the number of ways to express n as the sum of two squares (i.e. the number of pairs (a,b) such that a^2 + b^2 = n.) For instance, r(5) = 8 (as shown on the midterm.)<br />
<br />
Problem B. Prove that, for any N, there exists an integer n such that r(n) > N. (I.E., the function r(n) is ''unbounded.''<br />
<br />
Problem C. If you like Sage, write a short program in Sage to compute r(n) and compute the average of r(n) as n ranges from 1 to 1000. Whether or not you like Sage, make a guess as to how this average would behave if you replaced "1000" by a larger and larger number. (Feel free to ask the Sage-lovers what answer they got in the optional first part of the question.) Can you prove this guess is correct?<br />
<br />
Problem D. We saw in class that the ring Z[sqrt(-5)] doesn't have unique factorization; 6 can be factored as 2*3 or (1+sqrt(-5))(1-sqrt(-5)). In this problem, we will prove that Z[sqrt(-d)] fails to have unique factorization for EVERY odd d >= 5. (Actually it's true for all d >= 5 but to make the proof manageable we'll restrict to the odd case.]<br />
<br />
D.1. Show that (1+sqrt(-d)), (1-sqrt(-d)) and 2 are irreducible in Z[sqrt(-d)].<br />
<br />
D.2. Now give an element of Z[sqrt(-d)] that has two distinct factorizations into irreducibles. (Hint: imitate the example we used for Z[sqrt(-5)].)<br />
''Remark:'' The rings Z[sqrt(d)], where d is positive, are quite different -- here we believe that there are infinitely many with unique factorization, though this conjecture has remained unproved for many decades!<br />
<br />
*'''November 16'''<br />
<br />
Book problems: 5.3,5.4,5.5<br />
<br />
Problem A: We discussed in class the problem of studying which positive integers are the sum of two squares. In this problem we prove that every element n of (Z/pZ) is the sum of two squares. We argue as follows. Let S be the set of squares in (Z/pZ), and let T be the set of (Z/pZ) consisting of all elements of the form n - x^2, for some x in (Z/pZ).<br />
<br />
A.1. What is the size of S and of T? Use this to show that S and T are not disjoint.<br />
<br />
A.2. Given that S and T are not disjoint, prove that n is the sum of two squares.<br />
<br />
Problem B. Using the continued fraction expansion, find a solution to the Pell equation x^2 - 13 y^2 = 1.<br />
<br />
Problem C. Show that the modified Pell equation x^2 - 7y^2 = -1 has no solutions in integers x,y. (Hint: reduce the equation modulo a suitably chosen prime.)<br />
<br />
Problem D. Stuffy Stirnweiss finished the 1945 season with a batting average of .3085443. Using continued fractions, guess how many at-bats he had. Tony Cuccinello had a batting average of .3084577. Given that he had more than 200 and fewer than 600 at-bats, can you estimate the number of at-bats he had?<br />
<br />
'''December 7'''<br />
<br />
Problem A. When p is a prime congruent to 3 mod 4, prove that ((p-1)/2)! is either 1 or -1 in (Z/pZ). '''OPTIONAL:''' Use sage to compute whether this factorial is 1 or -1 for many primes p. Is there any pattern? Does it seem to be 1 half the time and -1 half the time? Note that I have no idea what the answer to this question is.<br />
<br />
Problem B. Using the continued fraction expansion, find a solution to the Pell equation x^2 - 13 y^2 = 1.<br />
<br />
Problem C. Show that the modified Pell equation x^2 - 7y^2 = -1 has no solutions in integers x,y. (Hint: reduce the equation modulo a suitably chosen prime.)<br />
<br />
Problem D.1. Pick two values of a in F_11 = Z/11Z (a not equal to 3), such that the equation y^2 = x^3+ax+1 defines an elliptic curve (i.e., it is smooth). For each such a, determine the number of points #E(F_11) and check that it falls inside the interval described in class. <br />
<br />
Problem D.2. The point P = (0,1) lies on each of these curves. For a =3 (the curve discussed in class) determine the order of P in the elliptic curve group, that is, find the smallest positive integer n such that nP = (infinity) -- the identity element in the group law.<br />
<br />
<!-- '''November 19'''<br />
<br />
Problem A. Using the method discussed in class (which is also the method of problem 4.6 in Stein, which in retrospect I think was too hard to assign with no preparation) find a nontrivial solution with y > 0 to the equation x^2 + 11y^2 = z^2. <br />
Problem B. Let f(X) be the number of solutions of A^2 + B^2 = C^3 such that A^2, B^2, and C^3 are all at most X. Use the heuristic described in class to explain why one might expect f(X) to grow more or less like X^{1/3}.<br />
<br />
Problem C. Show that if A+Bi is the cube of a Gaussian integer, then A^2 + B^2 is a perfect cube.<br />
<br />
Do ONE of problem D1 and D2; D1 is for people who like Sage, D2 is for people who like proving things.<br />
<br />
D1. Use Sage to compute f(X) for X = 1000, 10000, 100000. Does the answer look consistent with the heuristic prediction you made in B? Does f(X)/X^{1/3} appear to be approaching a limit?<br />
D2. We will use problem C to give a lower bound for f(X). Use this to show that there are at least X^{1/3} Gaussian integers with norm at most X that are perfect cubes. From here, show that f(X) > X^{1/3}.<br />
<br />
OPTIONAL: Show that the converse of problem C also holds -- A+Bi has norm a perfect cube if and only if A+Bi is a perfect cube. (You will need unique factorization of Gaussian integers.) Using this fact, prove that f(X) is bounded above by C X^{1/3} for some constant C.<br />
<br />
'''December 8 (note nonstandard due date)'''<br />
<br />
Problem A: Hyperbolas, ellipses, and "magic slopes"<br />
<br />
In the usual analytic geometry, both hyperbolas and ellipses are given by equations of the form<br />
<br />
ax^2 + bxy + cy^2 = d (*)<br />
<br />
<br />
(we always assume d is nonzero.)<br />
<br />
How can we tell whether such an equation describes an ellipse or a<br />
hyperbola? Well, in the geometric setting we all know and love (i.e. in<br />
R^2) a hyperbola has asymptotes and an ellipse does not. What is an <br />
asymptote? You could say it's "a line which doesn't intersect the curve", <br />
but ellipses have such lines too. You might want to say "a line which <br />
doesn't intersect the curve but comes closer and closer to the curve," but <br />
the problem here is that this a) is somewhat imprecise, and b) relies on a <br />
notion of "closer" that is not going to be very clear in Z/pZ, which is <br />
our ultimate goal!<br />
<br />
<br />
So let me state it a different way: let's say the asymptotes of a <br />
hyperbola have slopes m_1 and m_2. These are "magic slopes" in the <br />
following sense: ANY line of slope m_1 (and ditto for m_2) intersects the <br />
hyperbola in at most one point. An ellipse doesn't have "magic slopes" -- <br />
you can convince yourself by drawing pictures that, for any slope m, you <br />
can find a line of slope m striking the ellipse twice.<br />
<br />
<br />
[REMARK: For technical reasons we're going to ignore hyperbolae with a <br />
vertical asymptote, since vertical lines don't exactly have slope.]<br />
<br />
<br />
Now you might know the criterion that, over the real numbers.<br />
<br />
<br />
ax^2 + bxy + cy^2 = d<br />
<br />
<br />
is a hyperbola if b^2 - 4ac > 0 and an ellipse if b^2 - 4ac < 0. (For <br />
instance, x^2 - y^2 = 1 is a hyperbola with magic slopes 1 and -1, and x^2 <br />
+ y^2 = 1 is an ellipse.) <br />
This criterion does not make sense in Z/pZ, where there is no notion of <br />
"greater" or "less."<br />
<br />
<br />
QUESTION A.1.<br />
<br />
For which primes p is it the case that <br />
<br />
x^2 + y^2 = 1<br />
<br />
has magic slopes in Z/pZ? <br />
<br />
QUESTION A.2. For which primes p is it the case that<br />
<br />
<br />
x^2 - y^2 = 1<br />
<br />
<br />
has magic slopes in Z/pZ?<br />
<br />
<br />
QUESTION B: Back in the very first week of this course we studied the set of integers x such that x^2 + 1 is prime. Based on the heuristics we discussed in class, about how many x between 1 and N would you expect to satisfy "x^2 + 1 is prime?"<br />
<br />
<br />
QUESTION C: Let z be a complex number. Then the set of all complex numbers of the form a + bz, with a and b integers, forms a lattice in the complex plane (thought of as R^2.) What is the covolume of this lattice, in terms of z?<br />
<br />
<br />
QUESTION D: Let z be a complex number. Using Minkowski's theorem and the result of question C, show that there exist integers a,b, not both zero, such that Norm(a+bz) is at most (4/pi) Im(z). (Hint: let Omega be the set of all complex numbers of norm at most (4/pi) Im(z).)<br />
<br />
<br />
OPTIONAL EXTRA: Can you give an exact formula for the minimal norm of any nonzero complex number of the form a+bz?<br />
<br />
'''December 12''' Book problems 6.1, 6.2, 6.5, 6.10. For extra credit attempt to do problem 6.9.<br />
<br />
--></div>Andreichttps://www.math.wisc.edu/wiki/index.php?title=Math_567_--_Elementary_Number_Theory&diff=19087Math 567 -- Elementary Number Theory2020-02-21T20:40:35Z<p>Andreic: </p>
<hr />
<div>'''MATH 567<br />
<br />
Elementary Number Theory'''<br />
<br />
MWF 1:20-2:10, Van Vleck B119<br />
<br />
'''Professor:''' [http://www.math.wisc.edu/~andreic/ Andrei Caldararu] (andreic@math.wisc.edu)<br />
<br />
''Office Hours:'' Wednesdays 2:30-3:30, Van Vleck 605. <br />
<br />
'''Grader:''' Yifan Peng, email: peng64@wisc.edu<br />
<br />
''Office Hours:'' TBA.<br />
<br />
Math 567 is a course in elementary number theory, aimed at undergraduates majoring in math or other quantitative disciplines. A general familiarity with abstract algebra at the level of Math 541 will be assumed, but students who haven't taken 541 are welcome to attend if they're willing to play a little catchup. We will be using William Stein's new (and cheap) textbook [http://www.amazon.com/Elementary-Number-Theory-Computational-Undergraduate/dp/0387855246 Elementary Number Theory: Primes, Congruences, and Secrets], which emphasizes computational approaches to the subject. If you don't need a physical copy of the book, [http://www.williamstein.org/ent/ it is available as a free legal .pdf.] We will be using the (free, public-domain) mathematical software [http://www.sagemath.org/ SAGE], developed largely by Stein, as an integral component of our coursework. There is a [http://doc.sagemath.org/pdf/en/tutorial/SageTutorial.pdf useful online tutorial.] You can download SAGE to your own computer or [http://www.sagenb.org use it online].<br />
<br />
Topics include some subset of, but are not limited to: Divisibility, the Euclidean algorithm and the GCD, linear Diophantine equations, prime numbers and uniqueness of factorization. Congruences, Chinese remainder theorem, Fermat's "little" theorem, Wilson's theorem, Euler's theorem and totient function, the RSA cryptosystem, Rabin's encryption scheme, Diffie-Hellman key exchange protocol. Number-theoretic functions, multiplicative functions, Möbius inversion. Primitive roots and indices. Quadratic reciprocity and the Legendre symbol. Perfect numbers, Mersenne primes, Fermat primes. Pythagorean triples, Special cases of Fermat's "last" theorem. Fibonacci numbers. Continued fractions. Distribution of primes, discussion of prime number theorem. Primality testing and factoring algorithms.<br />
<br />
'''Course Policies:''' Homework will be due on Fridays. It can be turned in late only with ''advance'' permission from your grader. It is acceptable to use calculators and computers on homework (indeed, some of it will require a computer) but calculators are not allowed during exams. You are encouraged to work together on homework, but writeups must be done individually.<br />
<br />
Many of the problems in this course will ask you to prove things. I expect proofs to be written in English sentences; the proofs in Stein's book are a good model for the level of verbosity I am looking for. <br />
<br />
'''Grading:''' The grade in Math 567 will be composed of 50% homework, 25% midterm, 25% final. The midterm will be on March 11, in class. The final exam will be on 5/8/2020, 7:45AM-9:45AM in room TBA.<br />
<br />
'''Syllabus:''' <br />
(This may change as we see what pace works well for the course. All section numbers refer to Stein's book.)<br />
<br />
* Jan 22-31: Prime numbers, prime factorizations, Euclidean algorithm and GCD (1.1-1.2)<br />
* Feb 3-7: The integers mod n, Euler's theorem, the phi function (2.1-2.2)<br />
* Feb 10-14: Modular exponentiation, primality testing, and primitive roots (2.4-2.5)<br />
* Feb 17-21: Public-key cryptography and RSA (3.1-3.4)<br />
* Feb 24-28: Rabin's algorithm (not in the book); algebraic numbers <br />
* Mar 2-6: Quadratic reciprocity (4.1-4.4)<br />
* Mar 9, Mar 13: Finite and infinite continued fractions (5.1-5.3)<br />
* Mar 11: ''Midterm exam''<br />
* Mar 23-30: Continued fractions and diophantine approximation (5.4-5.5)<br />
* Apr 1-3: Diophantine equations I: Pell's equation and Lagrange's theorem<br />
* Apr 6-10: More diophantine equations, elliptic curves (6.1)<br />
* Apr 13-17: Applications of elliptic curves (6.2-6.3)<br />
* Apr 20-24: More applications of elliptic curves (6.4)<br />
* Apr 27-May 1: advanced topic TBD: maybe additional discussion of cryptographic techniques?<br />
<br />
'''Homework:'''<br />
Homework is due at the beginning of class on the specified Friday. Typing your homework is not a requirement, but if you don't already know LaTeX I highly recommend that you learn it and use it to typeset your homework. I will sometimes assign extra problems, which I will e-mail to the class list and include here.<br />
<br />
* '''Jan 31''': 1.1, 1.3, 1.5, 1.7 (use SAGE), 1.8, 1.14.<br />
Problem A: Use SAGE to compute the number of x in [1..N] such that x^2 + 1 is prime, for N = 100, N = 1000, and N = 10000. Let f(N) be the number of such x. <br />
<br />
a) Can you formulate a conjecture about the relationship between f(N) and N/log N? <br />
<br />
b) What if x^2 + 1 is replaced with x^2 + 2? Can you explain why x^2 + 2 appears less likely to be prime? (Hint: consider x mod 3.)<br />
<br />
c) Prove that f(N) is at most (1/2)N+1. (Hint: consider x mod 2.) <br />
<br />
d) Give as good an upper bound as you can for f(N).<br />
<br />
Note that, despite the evident regularities you'll observe in this problem, we do not even know whether there are infinitely many primes of the form x^2 + 1! You would become very famous if you proved this.<br />
<br />
* '''Feb 7''': 2.6 (the formulation of numerical evidence should be done by Sage if you've got Sage working, and by calculator if not; you can use an online tool like [http://primes.utm.edu/curios/includes/primetest.php this] to test whether a number is prime.) 2.8,2.9,2.11,2.12,2.14,2.19<br />
<br />
* '''Feb 17''': 2.15, 2.18, 2.23, 2.26, 2.30.<br />
<br />
Problem A: Prove that if n=pq, with p,q prime, then n is not a Carmichael number.<br />
<br />
* '''Feb 21''': Book problems: 3.4, 3.5, 3.6<br />
<br />
Problem A. Prove that there are infinitely many primes p such that 2 is '''not''' a primitive root in Z/pZ. We break this up into steps.<br />
<br />
Problem A.1. Prove that, if x is an element of Z/nZ, then x^2 is not a primitive root.<br />
<br />
Problem A.2. Prove that there are infinitely many primes p such that 2 is a square in Z/pZ. Hint: suppose there are only finitely many such primes p_1, .. p_r, and define N = (p_1 .. p_r)^2 - 2. Where can you go from here...?<br />
<br />
Problem A.3. Give a list of five primes p such that 2 is not a primitive root in Z/pZ (you can use the method of this proof or any other.)<br />
<br />
Problem B. Prove that 24 is the largest integer n such that every element of (Z/nZ)^* is a root of x^2-1.<br />
<br />
* '''Feb 28''': Problem A. Using p = 23 and q=31, show how to encrypt the message x=240 with Rabin's algorithm. Find all possible decryptions of your encrypted message.<br />
<br />
<!--<br />
<br />
Problem B. Give a prime factorization of the Gaussian integer 7+9i.<br />
<br />
Problem C. Read the notes from [http://www.math.ucsd.edu/~jverstra/Gaussian1.pdf here]. Note that Z[i] satisfies a ''reduction theorem'': if n and d are Gaussian integers, then there exists integers q and r such that n = qd + r and Norm(r) < Norm(d). But (by contrast with the case of Z) this d may not be unique. In some contexts it is better to be able to choose r uniquely, even if this means letting r have norm greater than Norm(d). <br />
<br />
C.1. When d = 1+2i, show that, for each n in Z[i], there is a '''unique''' pair (q,r) in Z[i] such that n = qd+r and r is contained in the set {0,1,2,3,4}. For instance, i can be written as i(1+2i) + 2, so we say i reduces to 2 mod (1+2i).<br />
(Hint: Suppose q(1+2i)+r = q'(1+2i) + r'. What can we say about (r-r'), and why is this incompatible with both r and r' being in {0,1,2,3,4}?<br />
<br />
C.2. Show that if n is an integer in Z, the reduction of n mod (1+2i) is equal to its reduction mod 5.<br />
<br />
Problem D. Let's try to figure out how to define "phi(d)" for a Gaussian integer d. Suppose S is a set of Gaussian integers such that every n in Z[i] can be written uniquely as qd+r, with q in Z[i] and r in S. (So for instance when d=1+2i, we showed in problem B that we can take S to be {0...4}. It would also be OK to take S to be {1..5} or {0,i,2i,1+i,1+2i}. In fact, it turns out that S has to be a set of size Norm(d) (I might or might not prove this in class; if not, feel free just to accept it.)<br />
<br />
Now define phi(d) to be the number of elements s of S such that s and d are coprime.<br />
<br />
D.1. Compute phi(1+2i) and phi(3).<br />
<br />
D.2. Prove that the value of phi(d) does not depend on the choice of S.<br />
<br />
D.3. Prove that every n in Z[i] which is coprime to 3 satisfies n^phi(3) = 1 mod 3; that is, Euler's theorem holds in this case. (You can prove this by direct computation; of course, if you want, you are welcome to prove that Euler's theorem holds for Z[i] in general, adapting the proof in Stein's book or the one we gave in class.)<br />
<br />
*'''Nov 2:''' 4.1 from the book.<br />
<br />
Problem A. Express 50005 as the sum of two squares.<br />
<br />
In the next two problems we denote by r(n) the number of ways to express n as the sum of two squares (i.e. the number of pairs (a,b) such that a^2 + b^2 = n.) For instance, r(5) = 8 (as shown on the midterm.)<br />
<br />
Problem B. Prove that, for any N, there exists an integer n such that r(n) > N. (I.E., the function r(n) is ''unbounded.''<br />
<br />
Problem C. If you like Sage, write a short program in Sage to compute r(n) and compute the average of r(n) as n ranges from 1 to 1000. Whether or not you like Sage, make a guess as to how this average would behave if you replaced "1000" by a larger and larger number. (Feel free to ask the Sage-lovers what answer they got in the optional first part of the question.) Can you prove this guess is correct?<br />
<br />
Problem D. We saw in class that the ring Z[sqrt(-5)] doesn't have unique factorization; 6 can be factored as 2*3 or (1+sqrt(-5))(1-sqrt(-5)). In this problem, we will prove that Z[sqrt(-d)] fails to have unique factorization for EVERY odd d >= 5. (Actually it's true for all d >= 5 but to make the proof manageable we'll restrict to the odd case.]<br />
<br />
D.1. Show that (1+sqrt(-d)), (1-sqrt(-d)) and 2 are irreducible in Z[sqrt(-d)].<br />
<br />
D.2. Now give an element of Z[sqrt(-d)] that has two distinct factorizations into irreducibles. (Hint: imitate the example we used for Z[sqrt(-5)].)<br />
''Remark:'' The rings Z[sqrt(d)], where d is positive, are quite different -- here we believe that there are infinitely many with unique factorization, though this conjecture has remained unproved for many decades!<br />
<br />
*'''November 16'''<br />
<br />
Book problems: 5.3,5.4,5.5<br />
<br />
Problem A: We discussed in class the problem of studying which positive integers are the sum of two squares. In this problem we prove that every element n of (Z/pZ) is the sum of two squares. We argue as follows. Let S be the set of squares in (Z/pZ), and let T be the set of (Z/pZ) consisting of all elements of the form n - x^2, for some x in (Z/pZ).<br />
<br />
A.1. What is the size of S and of T? Use this to show that S and T are not disjoint.<br />
<br />
A.2. Given that S and T are not disjoint, prove that n is the sum of two squares.<br />
<br />
Problem B. Using the continued fraction expansion, find a solution to the Pell equation x^2 - 13 y^2 = 1.<br />
<br />
Problem C. Show that the modified Pell equation x^2 - 7y^2 = -1 has no solutions in integers x,y. (Hint: reduce the equation modulo a suitably chosen prime.)<br />
<br />
Problem D. Stuffy Stirnweiss finished the 1945 season with a batting average of .3085443. Using continued fractions, guess how many at-bats he had. Tony Cuccinello had a batting average of .3084577. Given that he had more than 200 and fewer than 600 at-bats, can you estimate the number of at-bats he had?<br />
<br />
'''December 7'''<br />
<br />
Problem A. When p is a prime congruent to 3 mod 4, prove that ((p-1)/2)! is either 1 or -1 in (Z/pZ). '''OPTIONAL:''' Use sage to compute whether this factorial is 1 or -1 for many primes p. Is there any pattern? Does it seem to be 1 half the time and -1 half the time? Note that I have no idea what the answer to this question is.<br />
<br />
Problem B. Using the continued fraction expansion, find a solution to the Pell equation x^2 - 13 y^2 = 1.<br />
<br />
Problem C. Show that the modified Pell equation x^2 - 7y^2 = -1 has no solutions in integers x,y. (Hint: reduce the equation modulo a suitably chosen prime.)<br />
<br />
Problem D.1. Pick two values of a in F_11 = Z/11Z (a not equal to 3), such that the equation y^2 = x^3+ax+1 defines an elliptic curve (i.e., it is smooth). For each such a, determine the number of points #E(F_11) and check that it falls inside the interval described in class. <br />
<br />
Problem D.2. The point P = (0,1) lies on each of these curves. For a =3 (the curve discussed in class) determine the order of P in the elliptic curve group, that is, find the smallest positive integer n such that nP = (infinity) -- the identity element in the group law.<br />
<br />
<!-- '''November 19'''<br />
<br />
Problem A. Using the method discussed in class (which is also the method of problem 4.6 in Stein, which in retrospect I think was too hard to assign with no preparation) find a nontrivial solution with y > 0 to the equation x^2 + 11y^2 = z^2. <br />
Problem B. Let f(X) be the number of solutions of A^2 + B^2 = C^3 such that A^2, B^2, and C^3 are all at most X. Use the heuristic described in class to explain why one might expect f(X) to grow more or less like X^{1/3}.<br />
<br />
Problem C. Show that if A+Bi is the cube of a Gaussian integer, then A^2 + B^2 is a perfect cube.<br />
<br />
Do ONE of problem D1 and D2; D1 is for people who like Sage, D2 is for people who like proving things.<br />
<br />
D1. Use Sage to compute f(X) for X = 1000, 10000, 100000. Does the answer look consistent with the heuristic prediction you made in B? Does f(X)/X^{1/3} appear to be approaching a limit?<br />
D2. We will use problem C to give a lower bound for f(X). Use this to show that there are at least X^{1/3} Gaussian integers with norm at most X that are perfect cubes. From here, show that f(X) > X^{1/3}.<br />
<br />
OPTIONAL: Show that the converse of problem C also holds -- A+Bi has norm a perfect cube if and only if A+Bi is a perfect cube. (You will need unique factorization of Gaussian integers.) Using this fact, prove that f(X) is bounded above by C X^{1/3} for some constant C.<br />
<br />
'''December 8 (note nonstandard due date)'''<br />
<br />
Problem A: Hyperbolas, ellipses, and "magic slopes"<br />
<br />
In the usual analytic geometry, both hyperbolas and ellipses are given by equations of the form<br />
<br />
ax^2 + bxy + cy^2 = d (*)<br />
<br />
<br />
(we always assume d is nonzero.)<br />
<br />
How can we tell whether such an equation describes an ellipse or a<br />
hyperbola? Well, in the geometric setting we all know and love (i.e. in<br />
R^2) a hyperbola has asymptotes and an ellipse does not. What is an <br />
asymptote? You could say it's "a line which doesn't intersect the curve", <br />
but ellipses have such lines too. You might want to say "a line which <br />
doesn't intersect the curve but comes closer and closer to the curve," but <br />
the problem here is that this a) is somewhat imprecise, and b) relies on a <br />
notion of "closer" that is not going to be very clear in Z/pZ, which is <br />
our ultimate goal!<br />
<br />
<br />
So let me state it a different way: let's say the asymptotes of a <br />
hyperbola have slopes m_1 and m_2. These are "magic slopes" in the <br />
following sense: ANY line of slope m_1 (and ditto for m_2) intersects the <br />
hyperbola in at most one point. An ellipse doesn't have "magic slopes" -- <br />
you can convince yourself by drawing pictures that, for any slope m, you <br />
can find a line of slope m striking the ellipse twice.<br />
<br />
<br />
[REMARK: For technical reasons we're going to ignore hyperbolae with a <br />
vertical asymptote, since vertical lines don't exactly have slope.]<br />
<br />
<br />
Now you might know the criterion that, over the real numbers.<br />
<br />
<br />
ax^2 + bxy + cy^2 = d<br />
<br />
<br />
is a hyperbola if b^2 - 4ac > 0 and an ellipse if b^2 - 4ac < 0. (For <br />
instance, x^2 - y^2 = 1 is a hyperbola with magic slopes 1 and -1, and x^2 <br />
+ y^2 = 1 is an ellipse.) <br />
This criterion does not make sense in Z/pZ, where there is no notion of <br />
"greater" or "less."<br />
<br />
<br />
QUESTION A.1.<br />
<br />
For which primes p is it the case that <br />
<br />
x^2 + y^2 = 1<br />
<br />
has magic slopes in Z/pZ? <br />
<br />
QUESTION A.2. For which primes p is it the case that<br />
<br />
<br />
x^2 - y^2 = 1<br />
<br />
<br />
has magic slopes in Z/pZ?<br />
<br />
<br />
QUESTION B: Back in the very first week of this course we studied the set of integers x such that x^2 + 1 is prime. Based on the heuristics we discussed in class, about how many x between 1 and N would you expect to satisfy "x^2 + 1 is prime?"<br />
<br />
<br />
QUESTION C: Let z be a complex number. Then the set of all complex numbers of the form a + bz, with a and b integers, forms a lattice in the complex plane (thought of as R^2.) What is the covolume of this lattice, in terms of z?<br />
<br />
<br />
QUESTION D: Let z be a complex number. Using Minkowski's theorem and the result of question C, show that there exist integers a,b, not both zero, such that Norm(a+bz) is at most (4/pi) Im(z). (Hint: let Omega be the set of all complex numbers of norm at most (4/pi) Im(z).)<br />
<br />
<br />
OPTIONAL EXTRA: Can you give an exact formula for the minimal norm of any nonzero complex number of the form a+bz?<br />
<br />
'''December 12''' Book problems 6.1, 6.2, 6.5, 6.10. For extra credit attempt to do problem 6.9.<br />
<br />
--></div>Andreichttps://www.math.wisc.edu/wiki/index.php?title=Math_567_--_Elementary_Number_Theory&diff=19046Math 567 -- Elementary Number Theory2020-02-17T20:32:26Z<p>Andreic: </p>
<hr />
<div>'''MATH 567<br />
<br />
Elementary Number Theory'''<br />
<br />
MWF 1:20-2:10, Van Vleck B119<br />
<br />
'''Professor:''' [http://www.math.wisc.edu/~andreic/ Andrei Caldararu] (andreic@math.wisc.edu)<br />
<br />
''Office Hours:'' Wednesdays 2:30-3:30, Van Vleck 605. <br />
<br />
'''Grader:''' Yifan Peng, email: peng64@wisc.edu<br />
<br />
''Office Hours:'' TBA.<br />
<br />
Math 567 is a course in elementary number theory, aimed at undergraduates majoring in math or other quantitative disciplines. A general familiarity with abstract algebra at the level of Math 541 will be assumed, but students who haven't taken 541 are welcome to attend if they're willing to play a little catchup. We will be using William Stein's new (and cheap) textbook [http://www.amazon.com/Elementary-Number-Theory-Computational-Undergraduate/dp/0387855246 Elementary Number Theory: Primes, Congruences, and Secrets], which emphasizes computational approaches to the subject. If you don't need a physical copy of the book, [http://www.williamstein.org/ent/ it is available as a free legal .pdf.] We will be using the (free, public-domain) mathematical software [http://www.sagemath.org/ SAGE], developed largely by Stein, as an integral component of our coursework. There is a [http://doc.sagemath.org/pdf/en/tutorial/SageTutorial.pdf useful online tutorial.] You can download SAGE to your own computer or [http://www.sagenb.org use it online].<br />
<br />
Topics include some subset of, but are not limited to: Divisibility, the Euclidean algorithm and the GCD, linear Diophantine equations, prime numbers and uniqueness of factorization. Congruences, Chinese remainder theorem, Fermat's "little" theorem, Wilson's theorem, Euler's theorem and totient function, the RSA cryptosystem, Rabin's encryption scheme, Diffie-Hellman key exchange protocol. Number-theoretic functions, multiplicative functions, Möbius inversion. Primitive roots and indices. Quadratic reciprocity and the Legendre symbol. Perfect numbers, Mersenne primes, Fermat primes. Pythagorean triples, Special cases of Fermat's "last" theorem. Fibonacci numbers. Continued fractions. Distribution of primes, discussion of prime number theorem. Primality testing and factoring algorithms.<br />
<br />
'''Course Policies:''' Homework will be due on Fridays. It can be turned in late only with ''advance'' permission from your grader. It is acceptable to use calculators and computers on homework (indeed, some of it will require a computer) but calculators are not allowed during exams. You are encouraged to work together on homework, but writeups must be done individually.<br />
<br />
Many of the problems in this course will ask you to prove things. I expect proofs to be written in English sentences; the proofs in Stein's book are a good model for the level of verbosity I am looking for. <br />
<br />
'''Grading:''' The grade in Math 567 will be composed of 50% homework, 25% midterm, 25% final. The midterm will be on March 11, in class. The final exam will be on 5/8/2020, 7:45AM-9:45AM in room TBA.<br />
<br />
'''Syllabus:''' <br />
(This may change as we see what pace works well for the course. All section numbers refer to Stein's book.)<br />
<br />
* Jan 22-31: Prime numbers, prime factorizations, Euclidean algorithm and GCD (1.1-1.2)<br />
* Feb 3-7: The integers mod n, Euler's theorem, the phi function (2.1-2.2)<br />
* Feb 10-14: Modular exponentiation, primality testing, and primitive roots (2.4-2.5)<br />
* Feb 17-21: Public-key cryptography and RSA (3.1-3.4)<br />
* Feb 24-28: Rabin's algorithm (not in the book); algebraic numbers <br />
* Mar 2-6: Quadratic reciprocity (4.1-4.4)<br />
* Mar 9, Mar 13: Finite and infinite continued fractions (5.1-5.3)<br />
* Mar 11: ''Midterm exam''<br />
* Mar 23-30: Continued fractions and diophantine approximation (5.4-5.5)<br />
* Apr 1-3: Diophantine equations I: Pell's equation and Lagrange's theorem<br />
* Apr 6-10: More diophantine equations, elliptic curves (6.1)<br />
* Apr 13-17: Applications of elliptic curves (6.2-6.3)<br />
* Apr 20-24: More applications of elliptic curves (6.4)<br />
* Apr 27-May 1: advanced topic TBD: maybe additional discussion of cryptographic techniques?<br />
<br />
'''Homework:'''<br />
Homework is due at the beginning of class on the specified Friday. Typing your homework is not a requirement, but if you don't already know LaTeX I highly recommend that you learn it and use it to typeset your homework. I will sometimes assign extra problems, which I will e-mail to the class list and include here.<br />
<br />
* '''Jan 31''': 1.1, 1.3, 1.5, 1.7 (use SAGE), 1.8, 1.14.<br />
Problem A: Use SAGE to compute the number of x in [1..N] such that x^2 + 1 is prime, for N = 100, N = 1000, and N = 10000. Let f(N) be the number of such x. <br />
<br />
a) Can you formulate a conjecture about the relationship between f(N) and N/log N? <br />
<br />
b) What if x^2 + 1 is replaced with x^2 + 2? Can you explain why x^2 + 2 appears less likely to be prime? (Hint: consider x mod 3.)<br />
<br />
c) Prove that f(N) is at most (1/2)N+1. (Hint: consider x mod 2.) <br />
<br />
d) Give as good an upper bound as you can for f(N).<br />
<br />
Note that, despite the evident regularities you'll observe in this problem, we do not even know whether there are infinitely many primes of the form x^2 + 1! You would become very famous if you proved this.<br />
<br />
* '''Feb 7''': 2.6 (the formulation of numerical evidence should be done by Sage if you've got Sage working, and by calculator if not; you can use an online tool like [http://primes.utm.edu/curios/includes/primetest.php this] to test whether a number is prime.) 2.8,2.9,2.11,2.12,2.14,2.19<br />
<br />
* '''Feb 17''': 2.15, 2.18, 2.23, 2.26, 2.30.<br />
<br />
Problem A: Prove that if n=pq, with p,q prime, then n is not a Carmichael number.<br />
<br />
* '''Feb 21''': Book problems: 3.4, 3.5, 3.6<br />
<br />
Problem A. Prove that there are infinitely many primes p such that 2 is '''not''' a primitive root in Z/pZ. We break this up into steps.<br />
<br />
Problem A.1. Prove that, if x is an element of Z/nZ, then x^2 is not a primitive root.<br />
<br />
Problem A.2. Prove that there are infinitely many primes p such that 2 is a square in Z/pZ. Hint: suppose there are only finitely many such primes p_1, .. p_r, and define N = (p_1 .. p_r)^2 - 2. Where can you go from here...?<br />
<br />
Problem A.3. Give a list of five primes p such that 2 is not a primitive root in Z/pZ (you can use the method of this proof or any other.)<br />
<br />
Problem B. Prove that 24 is the largest integer n such that every element of (Z/nZ)^* is a root of x^2-1.<br />
<br />
<!--<br />
* '''Oct 19''': Problem A. Using p = 23 and q=31, show how to encrypt the message x=240 with Rabin's algorithm. Find all possible decryptions of your encrypted message.<br />
<br />
Problem B. Give a prime factorization of the Gaussian integer 7+9i.<br />
<br />
Problem C. Read the notes from [http://www.math.ucsd.edu/~jverstra/Gaussian1.pdf here]. Note that Z[i] satisfies a ''reduction theorem'': if n and d are Gaussian integers, then there exists integers q and r such that n = qd + r and Norm(r) < Norm(d). But (by contrast with the case of Z) this d may not be unique. In some contexts it is better to be able to choose r uniquely, even if this means letting r have norm greater than Norm(d). <br />
<br />
C.1. When d = 1+2i, show that, for each n in Z[i], there is a '''unique''' pair (q,r) in Z[i] such that n = qd+r and r is contained in the set {0,1,2,3,4}. For instance, i can be written as i(1+2i) + 2, so we say i reduces to 2 mod (1+2i).<br />
(Hint: Suppose q(1+2i)+r = q'(1+2i) + r'. What can we say about (r-r'), and why is this incompatible with both r and r' being in {0,1,2,3,4}?<br />
<br />
C.2. Show that if n is an integer in Z, the reduction of n mod (1+2i) is equal to its reduction mod 5.<br />
<br />
Problem D. Let's try to figure out how to define "phi(d)" for a Gaussian integer d. Suppose S is a set of Gaussian integers such that every n in Z[i] can be written uniquely as qd+r, with q in Z[i] and r in S. (So for instance when d=1+2i, we showed in problem B that we can take S to be {0...4}. It would also be OK to take S to be {1..5} or {0,i,2i,1+i,1+2i}. In fact, it turns out that S has to be a set of size Norm(d) (I might or might not prove this in class; if not, feel free just to accept it.)<br />
<br />
Now define phi(d) to be the number of elements s of S such that s and d are coprime.<br />
<br />
D.1. Compute phi(1+2i) and phi(3).<br />
<br />
D.2. Prove that the value of phi(d) does not depend on the choice of S.<br />
<br />
D.3. Prove that every n in Z[i] which is coprime to 3 satisfies n^phi(3) = 1 mod 3; that is, Euler's theorem holds in this case. (You can prove this by direct computation; of course, if you want, you are welcome to prove that Euler's theorem holds for Z[i] in general, adapting the proof in Stein's book or the one we gave in class.)<br />
<br />
*'''Nov 2:''' 4.1 from the book.<br />
<br />
Problem A. Express 50005 as the sum of two squares.<br />
<br />
In the next two problems we denote by r(n) the number of ways to express n as the sum of two squares (i.e. the number of pairs (a,b) such that a^2 + b^2 = n.) For instance, r(5) = 8 (as shown on the midterm.)<br />
<br />
Problem B. Prove that, for any N, there exists an integer n such that r(n) > N. (I.E., the function r(n) is ''unbounded.''<br />
<br />
Problem C. If you like Sage, write a short program in Sage to compute r(n) and compute the average of r(n) as n ranges from 1 to 1000. Whether or not you like Sage, make a guess as to how this average would behave if you replaced "1000" by a larger and larger number. (Feel free to ask the Sage-lovers what answer they got in the optional first part of the question.) Can you prove this guess is correct?<br />
<br />
Problem D. We saw in class that the ring Z[sqrt(-5)] doesn't have unique factorization; 6 can be factored as 2*3 or (1+sqrt(-5))(1-sqrt(-5)). In this problem, we will prove that Z[sqrt(-d)] fails to have unique factorization for EVERY odd d >= 5. (Actually it's true for all d >= 5 but to make the proof manageable we'll restrict to the odd case.]<br />
<br />
D.1. Show that (1+sqrt(-d)), (1-sqrt(-d)) and 2 are irreducible in Z[sqrt(-d)].<br />
<br />
D.2. Now give an element of Z[sqrt(-d)] that has two distinct factorizations into irreducibles. (Hint: imitate the example we used for Z[sqrt(-5)].)<br />
''Remark:'' The rings Z[sqrt(d)], where d is positive, are quite different -- here we believe that there are infinitely many with unique factorization, though this conjecture has remained unproved for many decades!<br />
<br />
*'''November 16'''<br />
<br />
Book problems: 5.3,5.4,5.5<br />
<br />
Problem A: We discussed in class the problem of studying which positive integers are the sum of two squares. In this problem we prove that every element n of (Z/pZ) is the sum of two squares. We argue as follows. Let S be the set of squares in (Z/pZ), and let T be the set of (Z/pZ) consisting of all elements of the form n - x^2, for some x in (Z/pZ).<br />
<br />
A.1. What is the size of S and of T? Use this to show that S and T are not disjoint.<br />
<br />
A.2. Given that S and T are not disjoint, prove that n is the sum of two squares.<br />
<br />
Problem B. Using the continued fraction expansion, find a solution to the Pell equation x^2 - 13 y^2 = 1.<br />
<br />
Problem C. Show that the modified Pell equation x^2 - 7y^2 = -1 has no solutions in integers x,y. (Hint: reduce the equation modulo a suitably chosen prime.)<br />
<br />
Problem D. Stuffy Stirnweiss finished the 1945 season with a batting average of .3085443. Using continued fractions, guess how many at-bats he had. Tony Cuccinello had a batting average of .3084577. Given that he had more than 200 and fewer than 600 at-bats, can you estimate the number of at-bats he had?<br />
<br />
'''December 7'''<br />
<br />
Problem A. When p is a prime congruent to 3 mod 4, prove that ((p-1)/2)! is either 1 or -1 in (Z/pZ). '''OPTIONAL:''' Use sage to compute whether this factorial is 1 or -1 for many primes p. Is there any pattern? Does it seem to be 1 half the time and -1 half the time? Note that I have no idea what the answer to this question is.<br />
<br />
Problem B. Using the continued fraction expansion, find a solution to the Pell equation x^2 - 13 y^2 = 1.<br />
<br />
Problem C. Show that the modified Pell equation x^2 - 7y^2 = -1 has no solutions in integers x,y. (Hint: reduce the equation modulo a suitably chosen prime.)<br />
<br />
Problem D.1. Pick two values of a in F_11 = Z/11Z (a not equal to 3), such that the equation y^2 = x^3+ax+1 defines an elliptic curve (i.e., it is smooth). For each such a, determine the number of points #E(F_11) and check that it falls inside the interval described in class. <br />
<br />
Problem D.2. The point P = (0,1) lies on each of these curves. For a =3 (the curve discussed in class) determine the order of P in the elliptic curve group, that is, find the smallest positive integer n such that nP = (infinity) -- the identity element in the group law.<br />
<br />
<!-- '''November 19'''<br />
<br />
Problem A. Using the method discussed in class (which is also the method of problem 4.6 in Stein, which in retrospect I think was too hard to assign with no preparation) find a nontrivial solution with y > 0 to the equation x^2 + 11y^2 = z^2. <br />
Problem B. Let f(X) be the number of solutions of A^2 + B^2 = C^3 such that A^2, B^2, and C^3 are all at most X. Use the heuristic described in class to explain why one might expect f(X) to grow more or less like X^{1/3}.<br />
<br />
Problem C. Show that if A+Bi is the cube of a Gaussian integer, then A^2 + B^2 is a perfect cube.<br />
<br />
Do ONE of problem D1 and D2; D1 is for people who like Sage, D2 is for people who like proving things.<br />
<br />
D1. Use Sage to compute f(X) for X = 1000, 10000, 100000. Does the answer look consistent with the heuristic prediction you made in B? Does f(X)/X^{1/3} appear to be approaching a limit?<br />
D2. We will use problem C to give a lower bound for f(X). Use this to show that there are at least X^{1/3} Gaussian integers with norm at most X that are perfect cubes. From here, show that f(X) > X^{1/3}.<br />
<br />
OPTIONAL: Show that the converse of problem C also holds -- A+Bi has norm a perfect cube if and only if A+Bi is a perfect cube. (You will need unique factorization of Gaussian integers.) Using this fact, prove that f(X) is bounded above by C X^{1/3} for some constant C.<br />
<br />
'''December 8 (note nonstandard due date)'''<br />
<br />
Problem A: Hyperbolas, ellipses, and "magic slopes"<br />
<br />
In the usual analytic geometry, both hyperbolas and ellipses are given by equations of the form<br />
<br />
ax^2 + bxy + cy^2 = d (*)<br />
<br />
<br />
(we always assume d is nonzero.)<br />
<br />
How can we tell whether such an equation describes an ellipse or a<br />
hyperbola? Well, in the geometric setting we all know and love (i.e. in<br />
R^2) a hyperbola has asymptotes and an ellipse does not. What is an <br />
asymptote? You could say it's "a line which doesn't intersect the curve", <br />
but ellipses have such lines too. You might want to say "a line which <br />
doesn't intersect the curve but comes closer and closer to the curve," but <br />
the problem here is that this a) is somewhat imprecise, and b) relies on a <br />
notion of "closer" that is not going to be very clear in Z/pZ, which is <br />
our ultimate goal!<br />
<br />
<br />
So let me state it a different way: let's say the asymptotes of a <br />
hyperbola have slopes m_1 and m_2. These are "magic slopes" in the <br />
following sense: ANY line of slope m_1 (and ditto for m_2) intersects the <br />
hyperbola in at most one point. An ellipse doesn't have "magic slopes" -- <br />
you can convince yourself by drawing pictures that, for any slope m, you <br />
can find a line of slope m striking the ellipse twice.<br />
<br />
<br />
[REMARK: For technical reasons we're going to ignore hyperbolae with a <br />
vertical asymptote, since vertical lines don't exactly have slope.]<br />
<br />
<br />
Now you might know the criterion that, over the real numbers.<br />
<br />
<br />
ax^2 + bxy + cy^2 = d<br />
<br />
<br />
is a hyperbola if b^2 - 4ac > 0 and an ellipse if b^2 - 4ac < 0. (For <br />
instance, x^2 - y^2 = 1 is a hyperbola with magic slopes 1 and -1, and x^2 <br />
+ y^2 = 1 is an ellipse.) <br />
This criterion does not make sense in Z/pZ, where there is no notion of <br />
"greater" or "less."<br />
<br />
<br />
QUESTION A.1.<br />
<br />
For which primes p is it the case that <br />
<br />
x^2 + y^2 = 1<br />
<br />
has magic slopes in Z/pZ? <br />
<br />
QUESTION A.2. For which primes p is it the case that<br />
<br />
<br />
x^2 - y^2 = 1<br />
<br />
<br />
has magic slopes in Z/pZ?<br />
<br />
<br />
QUESTION B: Back in the very first week of this course we studied the set of integers x such that x^2 + 1 is prime. Based on the heuristics we discussed in class, about how many x between 1 and N would you expect to satisfy "x^2 + 1 is prime?"<br />
<br />
<br />
QUESTION C: Let z be a complex number. Then the set of all complex numbers of the form a + bz, with a and b integers, forms a lattice in the complex plane (thought of as R^2.) What is the covolume of this lattice, in terms of z?<br />
<br />
<br />
QUESTION D: Let z be a complex number. Using Minkowski's theorem and the result of question C, show that there exist integers a,b, not both zero, such that Norm(a+bz) is at most (4/pi) Im(z). (Hint: let Omega be the set of all complex numbers of norm at most (4/pi) Im(z).)<br />
<br />
<br />
OPTIONAL EXTRA: Can you give an exact formula for the minimal norm of any nonzero complex number of the form a+bz?<br />
<br />
'''December 12''' Book problems 6.1, 6.2, 6.5, 6.10. For extra credit attempt to do problem 6.9.<br />
<br />
--></div>Andreichttps://www.math.wisc.edu/wiki/index.php?title=Math_567_--_Elementary_Number_Theory&diff=19045Math 567 -- Elementary Number Theory2020-02-17T20:32:06Z<p>Andreic: </p>
<hr />
<div>'''MATH 567<br />
<br />
Elementary Number Theory'''<br />
<br />
MWF 1:20-2:10, Van Vleck B119<br />
<br />
'''Professor:''' [http://www.math.wisc.edu/~andreic/ Andrei Caldararu] (andreic@math.wisc.edu)<br />
''Office Hours:'' Wednesdays 2:30-3:30, Van Vleck 605. <br />
<br />
'''Grader:''' Yifan Peng, email: peng64@wisc.edu<br />
''Office Hours:'' TBA.<br />
<br />
Math 567 is a course in elementary number theory, aimed at undergraduates majoring in math or other quantitative disciplines. A general familiarity with abstract algebra at the level of Math 541 will be assumed, but students who haven't taken 541 are welcome to attend if they're willing to play a little catchup. We will be using William Stein's new (and cheap) textbook [http://www.amazon.com/Elementary-Number-Theory-Computational-Undergraduate/dp/0387855246 Elementary Number Theory: Primes, Congruences, and Secrets], which emphasizes computational approaches to the subject. If you don't need a physical copy of the book, [http://www.williamstein.org/ent/ it is available as a free legal .pdf.] We will be using the (free, public-domain) mathematical software [http://www.sagemath.org/ SAGE], developed largely by Stein, as an integral component of our coursework. There is a [http://doc.sagemath.org/pdf/en/tutorial/SageTutorial.pdf useful online tutorial.] You can download SAGE to your own computer or [http://www.sagenb.org use it online].<br />
<br />
Topics include some subset of, but are not limited to: Divisibility, the Euclidean algorithm and the GCD, linear Diophantine equations, prime numbers and uniqueness of factorization. Congruences, Chinese remainder theorem, Fermat's "little" theorem, Wilson's theorem, Euler's theorem and totient function, the RSA cryptosystem, Rabin's encryption scheme, Diffie-Hellman key exchange protocol. Number-theoretic functions, multiplicative functions, Möbius inversion. Primitive roots and indices. Quadratic reciprocity and the Legendre symbol. Perfect numbers, Mersenne primes, Fermat primes. Pythagorean triples, Special cases of Fermat's "last" theorem. Fibonacci numbers. Continued fractions. Distribution of primes, discussion of prime number theorem. Primality testing and factoring algorithms.<br />
<br />
'''Course Policies:''' Homework will be due on Fridays. It can be turned in late only with ''advance'' permission from your grader. It is acceptable to use calculators and computers on homework (indeed, some of it will require a computer) but calculators are not allowed during exams. You are encouraged to work together on homework, but writeups must be done individually.<br />
<br />
Many of the problems in this course will ask you to prove things. I expect proofs to be written in English sentences; the proofs in Stein's book are a good model for the level of verbosity I am looking for. <br />
<br />
'''Grading:''' The grade in Math 567 will be composed of 50% homework, 25% midterm, 25% final. The midterm will be on March 11, in class. The final exam will be on 5/8/2020, 7:45AM-9:45AM in room TBA.<br />
<br />
'''Syllabus:''' <br />
(This may change as we see what pace works well for the course. All section numbers refer to Stein's book.)<br />
<br />
* Jan 22-31: Prime numbers, prime factorizations, Euclidean algorithm and GCD (1.1-1.2)<br />
* Feb 3-7: The integers mod n, Euler's theorem, the phi function (2.1-2.2)<br />
* Feb 10-14: Modular exponentiation, primality testing, and primitive roots (2.4-2.5)<br />
* Feb 17-21: Public-key cryptography and RSA (3.1-3.4)<br />
* Feb 24-28: Rabin's algorithm (not in the book); algebraic numbers <br />
* Mar 2-6: Quadratic reciprocity (4.1-4.4)<br />
* Mar 9, Mar 13: Finite and infinite continued fractions (5.1-5.3)<br />
* Mar 11: ''Midterm exam''<br />
* Mar 23-30: Continued fractions and diophantine approximation (5.4-5.5)<br />
* Apr 1-3: Diophantine equations I: Pell's equation and Lagrange's theorem<br />
* Apr 6-10: More diophantine equations, elliptic curves (6.1)<br />
* Apr 13-17: Applications of elliptic curves (6.2-6.3)<br />
* Apr 20-24: More applications of elliptic curves (6.4)<br />
* Apr 27-May 1: advanced topic TBD: maybe additional discussion of cryptographic techniques?<br />
<br />
'''Homework:'''<br />
Homework is due at the beginning of class on the specified Friday. Typing your homework is not a requirement, but if you don't already know LaTeX I highly recommend that you learn it and use it to typeset your homework. I will sometimes assign extra problems, which I will e-mail to the class list and include here.<br />
<br />
* '''Jan 31''': 1.1, 1.3, 1.5, 1.7 (use SAGE), 1.8, 1.14.<br />
Problem A: Use SAGE to compute the number of x in [1..N] such that x^2 + 1 is prime, for N = 100, N = 1000, and N = 10000. Let f(N) be the number of such x. <br />
<br />
a) Can you formulate a conjecture about the relationship between f(N) and N/log N? <br />
<br />
b) What if x^2 + 1 is replaced with x^2 + 2? Can you explain why x^2 + 2 appears less likely to be prime? (Hint: consider x mod 3.)<br />
<br />
c) Prove that f(N) is at most (1/2)N+1. (Hint: consider x mod 2.) <br />
<br />
d) Give as good an upper bound as you can for f(N).<br />
<br />
Note that, despite the evident regularities you'll observe in this problem, we do not even know whether there are infinitely many primes of the form x^2 + 1! You would become very famous if you proved this.<br />
<br />
* '''Feb 7''': 2.6 (the formulation of numerical evidence should be done by Sage if you've got Sage working, and by calculator if not; you can use an online tool like [http://primes.utm.edu/curios/includes/primetest.php this] to test whether a number is prime.) 2.8,2.9,2.11,2.12,2.14,2.19<br />
<br />
* '''Feb 17''': 2.15, 2.18, 2.23, 2.26, 2.30.<br />
<br />
Problem A: Prove that if n=pq, with p,q prime, then n is not a Carmichael number.<br />
<br />
* '''Feb 21''': Book problems: 3.4, 3.5, 3.6<br />
<br />
Problem A. Prove that there are infinitely many primes p such that 2 is '''not''' a primitive root in Z/pZ. We break this up into steps.<br />
<br />
Problem A.1. Prove that, if x is an element of Z/nZ, then x^2 is not a primitive root.<br />
<br />
Problem A.2. Prove that there are infinitely many primes p such that 2 is a square in Z/pZ. Hint: suppose there are only finitely many such primes p_1, .. p_r, and define N = (p_1 .. p_r)^2 - 2. Where can you go from here...?<br />
<br />
Problem A.3. Give a list of five primes p such that 2 is not a primitive root in Z/pZ (you can use the method of this proof or any other.)<br />
<br />
Problem B. Prove that 24 is the largest integer n such that every element of (Z/nZ)^* is a root of x^2-1.<br />
<br />
<!--<br />
* '''Oct 19''': Problem A. Using p = 23 and q=31, show how to encrypt the message x=240 with Rabin's algorithm. Find all possible decryptions of your encrypted message.<br />
<br />
Problem B. Give a prime factorization of the Gaussian integer 7+9i.<br />
<br />
Problem C. Read the notes from [http://www.math.ucsd.edu/~jverstra/Gaussian1.pdf here]. Note that Z[i] satisfies a ''reduction theorem'': if n and d are Gaussian integers, then there exists integers q and r such that n = qd + r and Norm(r) < Norm(d). But (by contrast with the case of Z) this d may not be unique. In some contexts it is better to be able to choose r uniquely, even if this means letting r have norm greater than Norm(d). <br />
<br />
C.1. When d = 1+2i, show that, for each n in Z[i], there is a '''unique''' pair (q,r) in Z[i] such that n = qd+r and r is contained in the set {0,1,2,3,4}. For instance, i can be written as i(1+2i) + 2, so we say i reduces to 2 mod (1+2i).<br />
(Hint: Suppose q(1+2i)+r = q'(1+2i) + r'. What can we say about (r-r'), and why is this incompatible with both r and r' being in {0,1,2,3,4}?<br />
<br />
C.2. Show that if n is an integer in Z, the reduction of n mod (1+2i) is equal to its reduction mod 5.<br />
<br />
Problem D. Let's try to figure out how to define "phi(d)" for a Gaussian integer d. Suppose S is a set of Gaussian integers such that every n in Z[i] can be written uniquely as qd+r, with q in Z[i] and r in S. (So for instance when d=1+2i, we showed in problem B that we can take S to be {0...4}. It would also be OK to take S to be {1..5} or {0,i,2i,1+i,1+2i}. In fact, it turns out that S has to be a set of size Norm(d) (I might or might not prove this in class; if not, feel free just to accept it.)<br />
<br />
Now define phi(d) to be the number of elements s of S such that s and d are coprime.<br />
<br />
D.1. Compute phi(1+2i) and phi(3).<br />
<br />
D.2. Prove that the value of phi(d) does not depend on the choice of S.<br />
<br />
D.3. Prove that every n in Z[i] which is coprime to 3 satisfies n^phi(3) = 1 mod 3; that is, Euler's theorem holds in this case. (You can prove this by direct computation; of course, if you want, you are welcome to prove that Euler's theorem holds for Z[i] in general, adapting the proof in Stein's book or the one we gave in class.)<br />
<br />
*'''Nov 2:''' 4.1 from the book.<br />
<br />
Problem A. Express 50005 as the sum of two squares.<br />
<br />
In the next two problems we denote by r(n) the number of ways to express n as the sum of two squares (i.e. the number of pairs (a,b) such that a^2 + b^2 = n.) For instance, r(5) = 8 (as shown on the midterm.)<br />
<br />
Problem B. Prove that, for any N, there exists an integer n such that r(n) > N. (I.E., the function r(n) is ''unbounded.''<br />
<br />
Problem C. If you like Sage, write a short program in Sage to compute r(n) and compute the average of r(n) as n ranges from 1 to 1000. Whether or not you like Sage, make a guess as to how this average would behave if you replaced "1000" by a larger and larger number. (Feel free to ask the Sage-lovers what answer they got in the optional first part of the question.) Can you prove this guess is correct?<br />
<br />
Problem D. We saw in class that the ring Z[sqrt(-5)] doesn't have unique factorization; 6 can be factored as 2*3 or (1+sqrt(-5))(1-sqrt(-5)). In this problem, we will prove that Z[sqrt(-d)] fails to have unique factorization for EVERY odd d >= 5. (Actually it's true for all d >= 5 but to make the proof manageable we'll restrict to the odd case.]<br />
<br />
D.1. Show that (1+sqrt(-d)), (1-sqrt(-d)) and 2 are irreducible in Z[sqrt(-d)].<br />
<br />
D.2. Now give an element of Z[sqrt(-d)] that has two distinct factorizations into irreducibles. (Hint: imitate the example we used for Z[sqrt(-5)].)<br />
''Remark:'' The rings Z[sqrt(d)], where d is positive, are quite different -- here we believe that there are infinitely many with unique factorization, though this conjecture has remained unproved for many decades!<br />
<br />
*'''November 16'''<br />
<br />
Book problems: 5.3,5.4,5.5<br />
<br />
Problem A: We discussed in class the problem of studying which positive integers are the sum of two squares. In this problem we prove that every element n of (Z/pZ) is the sum of two squares. We argue as follows. Let S be the set of squares in (Z/pZ), and let T be the set of (Z/pZ) consisting of all elements of the form n - x^2, for some x in (Z/pZ).<br />
<br />
A.1. What is the size of S and of T? Use this to show that S and T are not disjoint.<br />
<br />
A.2. Given that S and T are not disjoint, prove that n is the sum of two squares.<br />
<br />
Problem B. Using the continued fraction expansion, find a solution to the Pell equation x^2 - 13 y^2 = 1.<br />
<br />
Problem C. Show that the modified Pell equation x^2 - 7y^2 = -1 has no solutions in integers x,y. (Hint: reduce the equation modulo a suitably chosen prime.)<br />
<br />
Problem D. Stuffy Stirnweiss finished the 1945 season with a batting average of .3085443. Using continued fractions, guess how many at-bats he had. Tony Cuccinello had a batting average of .3084577. Given that he had more than 200 and fewer than 600 at-bats, can you estimate the number of at-bats he had?<br />
<br />
'''December 7'''<br />
<br />
Problem A. When p is a prime congruent to 3 mod 4, prove that ((p-1)/2)! is either 1 or -1 in (Z/pZ). '''OPTIONAL:''' Use sage to compute whether this factorial is 1 or -1 for many primes p. Is there any pattern? Does it seem to be 1 half the time and -1 half the time? Note that I have no idea what the answer to this question is.<br />
<br />
Problem B. Using the continued fraction expansion, find a solution to the Pell equation x^2 - 13 y^2 = 1.<br />
<br />
Problem C. Show that the modified Pell equation x^2 - 7y^2 = -1 has no solutions in integers x,y. (Hint: reduce the equation modulo a suitably chosen prime.)<br />
<br />
Problem D.1. Pick two values of a in F_11 = Z/11Z (a not equal to 3), such that the equation y^2 = x^3+ax+1 defines an elliptic curve (i.e., it is smooth). For each such a, determine the number of points #E(F_11) and check that it falls inside the interval described in class. <br />
<br />
Problem D.2. The point P = (0,1) lies on each of these curves. For a =3 (the curve discussed in class) determine the order of P in the elliptic curve group, that is, find the smallest positive integer n such that nP = (infinity) -- the identity element in the group law.<br />
<br />
<!-- '''November 19'''<br />
<br />
Problem A. Using the method discussed in class (which is also the method of problem 4.6 in Stein, which in retrospect I think was too hard to assign with no preparation) find a nontrivial solution with y > 0 to the equation x^2 + 11y^2 = z^2. <br />
Problem B. Let f(X) be the number of solutions of A^2 + B^2 = C^3 such that A^2, B^2, and C^3 are all at most X. Use the heuristic described in class to explain why one might expect f(X) to grow more or less like X^{1/3}.<br />
<br />
Problem C. Show that if A+Bi is the cube of a Gaussian integer, then A^2 + B^2 is a perfect cube.<br />
<br />
Do ONE of problem D1 and D2; D1 is for people who like Sage, D2 is for people who like proving things.<br />
<br />
D1. Use Sage to compute f(X) for X = 1000, 10000, 100000. Does the answer look consistent with the heuristic prediction you made in B? Does f(X)/X^{1/3} appear to be approaching a limit?<br />
D2. We will use problem C to give a lower bound for f(X). Use this to show that there are at least X^{1/3} Gaussian integers with norm at most X that are perfect cubes. From here, show that f(X) > X^{1/3}.<br />
<br />
OPTIONAL: Show that the converse of problem C also holds -- A+Bi has norm a perfect cube if and only if A+Bi is a perfect cube. (You will need unique factorization of Gaussian integers.) Using this fact, prove that f(X) is bounded above by C X^{1/3} for some constant C.<br />
<br />
'''December 8 (note nonstandard due date)'''<br />
<br />
Problem A: Hyperbolas, ellipses, and "magic slopes"<br />
<br />
In the usual analytic geometry, both hyperbolas and ellipses are given by equations of the form<br />
<br />
ax^2 + bxy + cy^2 = d (*)<br />
<br />
<br />
(we always assume d is nonzero.)<br />
<br />
How can we tell whether such an equation describes an ellipse or a<br />
hyperbola? Well, in the geometric setting we all know and love (i.e. in<br />
R^2) a hyperbola has asymptotes and an ellipse does not. What is an <br />
asymptote? You could say it's "a line which doesn't intersect the curve", <br />
but ellipses have such lines too. You might want to say "a line which <br />
doesn't intersect the curve but comes closer and closer to the curve," but <br />
the problem here is that this a) is somewhat imprecise, and b) relies on a <br />
notion of "closer" that is not going to be very clear in Z/pZ, which is <br />
our ultimate goal!<br />
<br />
<br />
So let me state it a different way: let's say the asymptotes of a <br />
hyperbola have slopes m_1 and m_2. These are "magic slopes" in the <br />
following sense: ANY line of slope m_1 (and ditto for m_2) intersects the <br />
hyperbola in at most one point. An ellipse doesn't have "magic slopes" -- <br />
you can convince yourself by drawing pictures that, for any slope m, you <br />
can find a line of slope m striking the ellipse twice.<br />
<br />
<br />
[REMARK: For technical reasons we're going to ignore hyperbolae with a <br />
vertical asymptote, since vertical lines don't exactly have slope.]<br />
<br />
<br />
Now you might know the criterion that, over the real numbers.<br />
<br />
<br />
ax^2 + bxy + cy^2 = d<br />
<br />
<br />
is a hyperbola if b^2 - 4ac > 0 and an ellipse if b^2 - 4ac < 0. (For <br />
instance, x^2 - y^2 = 1 is a hyperbola with magic slopes 1 and -1, and x^2 <br />
+ y^2 = 1 is an ellipse.) <br />
This criterion does not make sense in Z/pZ, where there is no notion of <br />
"greater" or "less."<br />
<br />
<br />
QUESTION A.1.<br />
<br />
For which primes p is it the case that <br />
<br />
x^2 + y^2 = 1<br />
<br />
has magic slopes in Z/pZ? <br />
<br />
QUESTION A.2. For which primes p is it the case that<br />
<br />
<br />
x^2 - y^2 = 1<br />
<br />
<br />
has magic slopes in Z/pZ?<br />
<br />
<br />
QUESTION B: Back in the very first week of this course we studied the set of integers x such that x^2 + 1 is prime. Based on the heuristics we discussed in class, about how many x between 1 and N would you expect to satisfy "x^2 + 1 is prime?"<br />
<br />
<br />
QUESTION C: Let z be a complex number. Then the set of all complex numbers of the form a + bz, with a and b integers, forms a lattice in the complex plane (thought of as R^2.) What is the covolume of this lattice, in terms of z?<br />
<br />
<br />
QUESTION D: Let z be a complex number. Using Minkowski's theorem and the result of question C, show that there exist integers a,b, not both zero, such that Norm(a+bz) is at most (4/pi) Im(z). (Hint: let Omega be the set of all complex numbers of norm at most (4/pi) Im(z).)<br />
<br />
<br />
OPTIONAL EXTRA: Can you give an exact formula for the minimal norm of any nonzero complex number of the form a+bz?<br />
<br />
'''December 12''' Book problems 6.1, 6.2, 6.5, 6.10. For extra credit attempt to do problem 6.9.<br />
<br />
--></div>Andreichttps://www.math.wisc.edu/wiki/index.php?title=Math_567_--_Elementary_Number_Theory&diff=19043Math 567 -- Elementary Number Theory2020-02-17T16:26:42Z<p>Andreic: </p>
<hr />
<div>'''MATH 567<br />
<br />
Elementary Number Theory'''<br />
<br />
MWF 1:20-2:10, Van Vleck B119<br />
<br />
'''Professor:''' [http://www.math.wisc.edu/~andreic/ Andrei Caldararu] (andreic@math.wisc.edu)<br />
''Office Hours:'' Wednesdays 2:30-3:30, Van Vleck 605. <br />
<br />
'''Grader:''' TBA<br />
''Office Hours:'' TBA.<br />
<br />
Math 567 is a course in elementary number theory, aimed at undergraduates majoring in math or other quantitative disciplines. A general familiarity with abstract algebra at the level of Math 541 will be assumed, but students who haven't taken 541 are welcome to attend if they're willing to play a little catchup. We will be using William Stein's new (and cheap) textbook [http://www.amazon.com/Elementary-Number-Theory-Computational-Undergraduate/dp/0387855246 Elementary Number Theory: Primes, Congruences, and Secrets], which emphasizes computational approaches to the subject. If you don't need a physical copy of the book, [http://www.williamstein.org/ent/ it is available as a free legal .pdf.] We will be using the (free, public-domain) mathematical software [http://www.sagemath.org/ SAGE], developed largely by Stein, as an integral component of our coursework. There is a [http://doc.sagemath.org/pdf/en/tutorial/SageTutorial.pdf useful online tutorial.] You can download SAGE to your own computer or [http://www.sagenb.org use it online].<br />
<br />
Topics include some subset of, but are not limited to: Divisibility, the Euclidean algorithm and the GCD, linear Diophantine equations, prime numbers and uniqueness of factorization. Congruences, Chinese remainder theorem, Fermat's "little" theorem, Wilson's theorem, Euler's theorem and totient function, the RSA cryptosystem, Rabin's encryption scheme, Diffie-Hellman key exchange protocol. Number-theoretic functions, multiplicative functions, Möbius inversion. Primitive roots and indices. Quadratic reciprocity and the Legendre symbol. Perfect numbers, Mersenne primes, Fermat primes. Pythagorean triples, Special cases of Fermat's "last" theorem. Fibonacci numbers. Continued fractions. Distribution of primes, discussion of prime number theorem. Primality testing and factoring algorithms.<br />
<br />
'''Course Policies:''' Homework will be due on Fridays. It can be turned in late only with ''advance'' permission from your grader. It is acceptable to use calculators and computers on homework (indeed, some of it will require a computer) but calculators are not allowed during exams. You are encouraged to work together on homework, but writeups must be done individually.<br />
<br />
Many of the problems in this course will ask you to prove things. I expect proofs to be written in English sentences; the proofs in Stein's book are a good model for the level of verbosity I am looking for. <br />
<br />
'''Grading:''' The grade in Math 567 will be composed of 50% homework, 25% midterm, 25% final. The midterm will be on March 11, in class. The final exam will be on 5/8/2020, 7:45AM-9:45AM in room TBA.<br />
<br />
'''Syllabus:''' <br />
(This may change as we see what pace works well for the course. All section numbers refer to Stein's book.)<br />
<br />
* Jan 22-31: Prime numbers, prime factorizations, Euclidean algorithm and GCD (1.1-1.2)<br />
* Feb 3-7: The integers mod n, Euler's theorem, the phi function (2.1-2.2)<br />
* Feb 10-14: Modular exponentiation, primality testing, and primitive roots (2.4-2.5)<br />
* Feb 17-21: Public-key cryptography and RSA (3.1-3.4)<br />
* Feb 24-28: Rabin's algorithm (not in the book); algebraic numbers <br />
* Mar 2-6: Quadratic reciprocity (4.1-4.4)<br />
* Mar 9, Mar 13: Finite and infinite continued fractions (5.1-5.3)<br />
* Mar 11: ''Midterm exam''<br />
* Mar 23-30: Continued fractions and diophantine approximation (5.4-5.5)<br />
* Apr 1-3: Diophantine equations I: Pell's equation and Lagrange's theorem<br />
* Apr 6-10: More diophantine equations, elliptic curves (6.1)<br />
* Apr 13-17: Applications of elliptic curves (6.2-6.3)<br />
* Apr 20-24: More applications of elliptic curves (6.4)<br />
* Apr 27-May 1: advanced topic TBD: maybe additional discussion of cryptographic techniques?<br />
<br />
'''Homework:'''<br />
Homework is due at the beginning of class on the specified Friday. Typing your homework is not a requirement, but if you don't already know LaTeX I highly recommend that you learn it and use it to typeset your homework. I will sometimes assign extra problems, which I will e-mail to the class list and include here.<br />
<br />
* '''Jan 31''': 1.1, 1.3, 1.5, 1.7 (use SAGE), 1.8, 1.14.<br />
Problem A: Use SAGE to compute the number of x in [1..N] such that x^2 + 1 is prime, for N = 100, N = 1000, and N = 10000. Let f(N) be the number of such x. <br />
<br />
a) Can you formulate a conjecture about the relationship between f(N) and N/log N? <br />
<br />
b) What if x^2 + 1 is replaced with x^2 + 2? Can you explain why x^2 + 2 appears less likely to be prime? (Hint: consider x mod 3.)<br />
<br />
c) Prove that f(N) is at most (1/2)N+1. (Hint: consider x mod 2.) <br />
<br />
d) Give as good an upper bound as you can for f(N).<br />
<br />
Note that, despite the evident regularities you'll observe in this problem, we do not even know whether there are infinitely many primes of the form x^2 + 1! You would become very famous if you proved this.<br />
<br />
* '''Feb 7''': 2.6 (the formulation of numerical evidence should be done by Sage if you've got Sage working, and by calculator if not; you can use an online tool like [http://primes.utm.edu/curios/includes/primetest.php this] to test whether a number is prime.) 2.8,2.9,2.11,2.12,2.14,2.19<br />
<br />
* '''Feb 17''': 2.15, 2.18, 2.23, 2.26, 2.30.<br />
<br />
Problem A: Prove that if n=pq, with p,q prime, then n is not a Carmichael number.<br />
<br />
* '''Feb 21''': Book problems: 3.4, 3.5, 3.6<br />
<br />
Problem A. Prove that there are infinitely many primes p such that 2 is '''not''' a primitive root in Z/pZ. We break this up into steps.<br />
<br />
Problem A.1. Prove that, if x is an element of Z/nZ, then x^2 is not a primitive root.<br />
<br />
Problem A.2. Prove that there are infinitely many primes p such that 2 is a square in Z/pZ. Hint: suppose there are only finitely many such primes p_1, .. p_r, and define N = (p_1 .. p_r)^2 - 2. Where can you go from here...?<br />
<br />
Problem A.3. Give a list of five primes p such that 2 is not a primitive root in Z/pZ (you can use the method of this proof or any other.)<br />
<br />
Problem B. Prove that 24 is the largest integer n such that every element of (Z/nZ)^* is a root of x^2-1.<br />
<br />
<!--<br />
* '''Oct 19''': Problem A. Using p = 23 and q=31, show how to encrypt the message x=240 with Rabin's algorithm. Find all possible decryptions of your encrypted message.<br />
<br />
Problem B. Give a prime factorization of the Gaussian integer 7+9i.<br />
<br />
Problem C. Read the notes from [http://www.math.ucsd.edu/~jverstra/Gaussian1.pdf here]. Note that Z[i] satisfies a ''reduction theorem'': if n and d are Gaussian integers, then there exists integers q and r such that n = qd + r and Norm(r) < Norm(d). But (by contrast with the case of Z) this d may not be unique. In some contexts it is better to be able to choose r uniquely, even if this means letting r have norm greater than Norm(d). <br />
<br />
C.1. When d = 1+2i, show that, for each n in Z[i], there is a '''unique''' pair (q,r) in Z[i] such that n = qd+r and r is contained in the set {0,1,2,3,4}. For instance, i can be written as i(1+2i) + 2, so we say i reduces to 2 mod (1+2i).<br />
(Hint: Suppose q(1+2i)+r = q'(1+2i) + r'. What can we say about (r-r'), and why is this incompatible with both r and r' being in {0,1,2,3,4}?<br />
<br />
C.2. Show that if n is an integer in Z, the reduction of n mod (1+2i) is equal to its reduction mod 5.<br />
<br />
Problem D. Let's try to figure out how to define "phi(d)" for a Gaussian integer d. Suppose S is a set of Gaussian integers such that every n in Z[i] can be written uniquely as qd+r, with q in Z[i] and r in S. (So for instance when d=1+2i, we showed in problem B that we can take S to be {0...4}. It would also be OK to take S to be {1..5} or {0,i,2i,1+i,1+2i}. In fact, it turns out that S has to be a set of size Norm(d) (I might or might not prove this in class; if not, feel free just to accept it.)<br />
<br />
Now define phi(d) to be the number of elements s of S such that s and d are coprime.<br />
<br />
D.1. Compute phi(1+2i) and phi(3).<br />
<br />
D.2. Prove that the value of phi(d) does not depend on the choice of S.<br />
<br />
D.3. Prove that every n in Z[i] which is coprime to 3 satisfies n^phi(3) = 1 mod 3; that is, Euler's theorem holds in this case. (You can prove this by direct computation; of course, if you want, you are welcome to prove that Euler's theorem holds for Z[i] in general, adapting the proof in Stein's book or the one we gave in class.)<br />
<br />
*'''Nov 2:''' 4.1 from the book.<br />
<br />
Problem A. Express 50005 as the sum of two squares.<br />
<br />
In the next two problems we denote by r(n) the number of ways to express n as the sum of two squares (i.e. the number of pairs (a,b) such that a^2 + b^2 = n.) For instance, r(5) = 8 (as shown on the midterm.)<br />
<br />
Problem B. Prove that, for any N, there exists an integer n such that r(n) > N. (I.E., the function r(n) is ''unbounded.''<br />
<br />
Problem C. If you like Sage, write a short program in Sage to compute r(n) and compute the average of r(n) as n ranges from 1 to 1000. Whether or not you like Sage, make a guess as to how this average would behave if you replaced "1000" by a larger and larger number. (Feel free to ask the Sage-lovers what answer they got in the optional first part of the question.) Can you prove this guess is correct?<br />
<br />
Problem D. We saw in class that the ring Z[sqrt(-5)] doesn't have unique factorization; 6 can be factored as 2*3 or (1+sqrt(-5))(1-sqrt(-5)). In this problem, we will prove that Z[sqrt(-d)] fails to have unique factorization for EVERY odd d >= 5. (Actually it's true for all d >= 5 but to make the proof manageable we'll restrict to the odd case.]<br />
<br />
D.1. Show that (1+sqrt(-d)), (1-sqrt(-d)) and 2 are irreducible in Z[sqrt(-d)].<br />
<br />
D.2. Now give an element of Z[sqrt(-d)] that has two distinct factorizations into irreducibles. (Hint: imitate the example we used for Z[sqrt(-5)].)<br />
''Remark:'' The rings Z[sqrt(d)], where d is positive, are quite different -- here we believe that there are infinitely many with unique factorization, though this conjecture has remained unproved for many decades!<br />
<br />
*'''November 16'''<br />
<br />
Book problems: 5.3,5.4,5.5<br />
<br />
Problem A: We discussed in class the problem of studying which positive integers are the sum of two squares. In this problem we prove that every element n of (Z/pZ) is the sum of two squares. We argue as follows. Let S be the set of squares in (Z/pZ), and let T be the set of (Z/pZ) consisting of all elements of the form n - x^2, for some x in (Z/pZ).<br />
<br />
A.1. What is the size of S and of T? Use this to show that S and T are not disjoint.<br />
<br />
A.2. Given that S and T are not disjoint, prove that n is the sum of two squares.<br />
<br />
Problem B. Using the continued fraction expansion, find a solution to the Pell equation x^2 - 13 y^2 = 1.<br />
<br />
Problem C. Show that the modified Pell equation x^2 - 7y^2 = -1 has no solutions in integers x,y. (Hint: reduce the equation modulo a suitably chosen prime.)<br />
<br />
Problem D. Stuffy Stirnweiss finished the 1945 season with a batting average of .3085443. Using continued fractions, guess how many at-bats he had. Tony Cuccinello had a batting average of .3084577. Given that he had more than 200 and fewer than 600 at-bats, can you estimate the number of at-bats he had?<br />
<br />
'''December 7'''<br />
<br />
Problem A. When p is a prime congruent to 3 mod 4, prove that ((p-1)/2)! is either 1 or -1 in (Z/pZ). '''OPTIONAL:''' Use sage to compute whether this factorial is 1 or -1 for many primes p. Is there any pattern? Does it seem to be 1 half the time and -1 half the time? Note that I have no idea what the answer to this question is.<br />
<br />
Problem B. Using the continued fraction expansion, find a solution to the Pell equation x^2 - 13 y^2 = 1.<br />
<br />
Problem C. Show that the modified Pell equation x^2 - 7y^2 = -1 has no solutions in integers x,y. (Hint: reduce the equation modulo a suitably chosen prime.)<br />
<br />
Problem D.1. Pick two values of a in F_11 = Z/11Z (a not equal to 3), such that the equation y^2 = x^3+ax+1 defines an elliptic curve (i.e., it is smooth). For each such a, determine the number of points #E(F_11) and check that it falls inside the interval described in class. <br />
<br />
Problem D.2. The point P = (0,1) lies on each of these curves. For a =3 (the curve discussed in class) determine the order of P in the elliptic curve group, that is, find the smallest positive integer n such that nP = (infinity) -- the identity element in the group law.<br />
<br />
<!-- '''November 19'''<br />
<br />
Problem A. Using the method discussed in class (which is also the method of problem 4.6 in Stein, which in retrospect I think was too hard to assign with no preparation) find a nontrivial solution with y > 0 to the equation x^2 + 11y^2 = z^2. <br />
Problem B. Let f(X) be the number of solutions of A^2 + B^2 = C^3 such that A^2, B^2, and C^3 are all at most X. Use the heuristic described in class to explain why one might expect f(X) to grow more or less like X^{1/3}.<br />
<br />
Problem C. Show that if A+Bi is the cube of a Gaussian integer, then A^2 + B^2 is a perfect cube.<br />
<br />
Do ONE of problem D1 and D2; D1 is for people who like Sage, D2 is for people who like proving things.<br />
<br />
D1. Use Sage to compute f(X) for X = 1000, 10000, 100000. Does the answer look consistent with the heuristic prediction you made in B? Does f(X)/X^{1/3} appear to be approaching a limit?<br />
D2. We will use problem C to give a lower bound for f(X). Use this to show that there are at least X^{1/3} Gaussian integers with norm at most X that are perfect cubes. From here, show that f(X) > X^{1/3}.<br />
<br />
OPTIONAL: Show that the converse of problem C also holds -- A+Bi has norm a perfect cube if and only if A+Bi is a perfect cube. (You will need unique factorization of Gaussian integers.) Using this fact, prove that f(X) is bounded above by C X^{1/3} for some constant C.<br />
<br />
'''December 8 (note nonstandard due date)'''<br />
<br />
Problem A: Hyperbolas, ellipses, and "magic slopes"<br />
<br />
In the usual analytic geometry, both hyperbolas and ellipses are given by equations of the form<br />
<br />
ax^2 + bxy + cy^2 = d (*)<br />
<br />
<br />
(we always assume d is nonzero.)<br />
<br />
How can we tell whether such an equation describes an ellipse or a<br />
hyperbola? Well, in the geometric setting we all know and love (i.e. in<br />
R^2) a hyperbola has asymptotes and an ellipse does not. What is an <br />
asymptote? You could say it's "a line which doesn't intersect the curve", <br />
but ellipses have such lines too. You might want to say "a line which <br />
doesn't intersect the curve but comes closer and closer to the curve," but <br />
the problem here is that this a) is somewhat imprecise, and b) relies on a <br />
notion of "closer" that is not going to be very clear in Z/pZ, which is <br />
our ultimate goal!<br />
<br />
<br />
So let me state it a different way: let's say the asymptotes of a <br />
hyperbola have slopes m_1 and m_2. These are "magic slopes" in the <br />
following sense: ANY line of slope m_1 (and ditto for m_2) intersects the <br />
hyperbola in at most one point. An ellipse doesn't have "magic slopes" -- <br />
you can convince yourself by drawing pictures that, for any slope m, you <br />
can find a line of slope m striking the ellipse twice.<br />
<br />
<br />
[REMARK: For technical reasons we're going to ignore hyperbolae with a <br />
vertical asymptote, since vertical lines don't exactly have slope.]<br />
<br />
<br />
Now you might know the criterion that, over the real numbers.<br />
<br />
<br />
ax^2 + bxy + cy^2 = d<br />
<br />
<br />
is a hyperbola if b^2 - 4ac > 0 and an ellipse if b^2 - 4ac < 0. (For <br />
instance, x^2 - y^2 = 1 is a hyperbola with magic slopes 1 and -1, and x^2 <br />
+ y^2 = 1 is an ellipse.) <br />
This criterion does not make sense in Z/pZ, where there is no notion of <br />
"greater" or "less."<br />
<br />
<br />
QUESTION A.1.<br />
<br />
For which primes p is it the case that <br />
<br />
x^2 + y^2 = 1<br />
<br />
has magic slopes in Z/pZ? <br />
<br />
QUESTION A.2. For which primes p is it the case that<br />
<br />
<br />
x^2 - y^2 = 1<br />
<br />
<br />
has magic slopes in Z/pZ?<br />
<br />
<br />
QUESTION B: Back in the very first week of this course we studied the set of integers x such that x^2 + 1 is prime. Based on the heuristics we discussed in class, about how many x between 1 and N would you expect to satisfy "x^2 + 1 is prime?"<br />
<br />
<br />
QUESTION C: Let z be a complex number. Then the set of all complex numbers of the form a + bz, with a and b integers, forms a lattice in the complex plane (thought of as R^2.) What is the covolume of this lattice, in terms of z?<br />
<br />
<br />
QUESTION D: Let z be a complex number. Using Minkowski's theorem and the result of question C, show that there exist integers a,b, not both zero, such that Norm(a+bz) is at most (4/pi) Im(z). (Hint: let Omega be the set of all complex numbers of norm at most (4/pi) Im(z).)<br />
<br />
<br />
OPTIONAL EXTRA: Can you give an exact formula for the minimal norm of any nonzero complex number of the form a+bz?<br />
<br />
'''December 12''' Book problems 6.1, 6.2, 6.5, 6.10. For extra credit attempt to do problem 6.9.<br />
<br />
--></div>Andreichttps://www.math.wisc.edu/wiki/index.php?title=Algebra_and_Algebraic_Geometry_Seminar_Spring_2020&diff=19031Algebra and Algebraic Geometry Seminar Spring 20202020-02-14T16:42:10Z<p>Andreic: </p>
<hr />
<div>== Spring 2020 Schedule ==<br />
<br />
{| cellpadding="8"<br />
!align="left" | date<br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s) <br />
|-<br />
|January 24<br />
|[http://www.math.ualberta.ca/~xichen// Xi Chen (Alberta)]<br />
|[[#Xi Chen|Rational Curves on K3 Surfaces]]<br />
|Michael K<br />
|-<br />
|January 31<br />
|[http://www.math.utah.edu/~letz// Janina Letz (Utah)]<br />
|[[#Janina Letz|Local to global principles for generation time over commutative rings]]<br />
|Daniel and Michael B<br />
|-<br />
|February 7<br />
|Jonathan Monta&#241;o (New Mexico State)<br />
|Asymptotic behavior of invariants of symbolic powers<br />
|Daniel<br />
|-<br />
|February 14<br />
|<br />
|<br />
| <br />
|-<br />
|February 21<br />
|Erika Ordog (Duke)<br />
|Minimal resolutions of monomial ideals<br />
|Daniel<br />
|-<br />
|February 28<br />
|<br />
|<br />
|<br />
|-<br />
|March 6<br />
|<br />
|<br />
|<br />
|-<br />
|March 13<br />
|Kevin Tucker (UIC)<br />
|TBD<br />
|Daniel<br />
|-<br />
|March 20<br />
|<br />
|<br />
|<br />
|-<br />
|March 27<br />
|[https://mcfaddin.github.io// Patrick McFaddin (Fordham)]<br />
|TBD<br />
|Michael B<br />
|-<br />
|March 30, 3:30pm '''(Monday, unusual date and time!)'''<br />
|[https://pages.uoregon.edu/honigs/ Katrina Honigs (Oregon)]<br />
|TBA<br />
|Andrei<br />
|-<br />
|April 3<br />
|<br />
| <br />
| <br />
|-<br />
|April 10<br />
|[https://sites.google.com/view/ruijie-yang/ Ruijie Yang (Stony Brook)]<br />
|TBD<br />
|Michael K<br />
|-<br />
|April 17<br />
|Remy van Dobben de Bruyn (Princeton/IAS)<br />
|TBD<br />
|Botong<br />
|-<br />
|May 1<br />
|Lazarsfeld Distinguished Lectures<br />
|<br />
|<br />
|-<br />
|May 8<br />
|<br />
|<br />
| <br />
|-<br />
|}<br />
<br />
== Abstracts ==<br />
===Xi Chen===<br />
'''Rational Curves on K3 Surfaces<br />
'''<br />
<br />
It is conjectured that there are infinitely many rational<br />
curves on every projective K3 surface. A large part of this conjecture<br />
was proved by Jun Li and Christian Liedtke, based on the<br />
characteristic p reduction method proposed by<br />
Bogomolov-Hassett-Tschinkel. They proved that there are infinitely<br />
many rational curves on every projective K3 surface of odd Picard<br />
rank. Over complex numbers, there are a few remaining cases: K3<br />
surfaces of Picard rank two excluding elliptic K3's and K3's with<br />
infinite automorphism groups and K3 surfaces with two particular<br />
Picard lattices of rank four. We have settled these leftover cases and also<br />
generalized the conjecture to the existence of curves of high genus.<br />
This is a joint work with Frank Gounelas and Christian Liedtke.<br />
<br />
===Janina Letz===<br />
'''Local to global principles for generation time over commutative rings<br />
'''<br />
<br />
Abstract: In the derived category of modules over a commutative<br />
noetherian ring a complex $G$ is said to generate a complex $X$ if the<br />
latter can be obtained from the former by taking finitely many summands<br />
and cones. The number of cones needed in this process is the generation<br />
time of $X$. In this talk I will present some local to global type<br />
results for computing this invariant, and also discuss some<br />
applications of these results.<br />
<br />
<br />
===Jonathan Montano===<br />
'''Asymptotic behavior of invariants of symbolic powers '''<br />
<br />
Abstract: The symbolic powers of an ideal is a filtration that encodes important algebraic and geometric information of the ideal and the variety it defines. Despite the importance and great results about symbolic powers, their complete structure is far from being understood. For example, we do not completely understand yet the behavior of the number of generators, regularities, and depths of these ideals. In this talk I will report on resent results in this direction in joint works with Hailong Dao and Luis Núñez-Betancourt.<br />
<br />
===Erika Ordog===<br />
'''Minimal resolutions of monomial ideals'''<br />
<br />
Abstract: The problem of finding minimal free resolutions of monomial<br />
ideals in polynomial rings has been central to commutative<br />
algebra ever since Kaplansky raised the problem in the 1960s and<br />
his student, Diana Taylor, produced the first general<br />
construction in 1966. The ultimate goal along these lines is a<br />
construction of free resolutions that is universal -- that is,<br />
valid for arbitrary monomial ideals -- canonical, combinatorial,<br />
and minimal. This talk describes a solution to the problem<br />
valid in characteristic 0 and almost all positive characteristics.<br />
<br />
<br />
===Kevin Tucker===<br />
'''An Analog of PLT Singularities in Mixed Characteristic'''<br />
<br />
Abstract: In algebraic geometry, singularities are often understood using hyperplane sections. For example, if the singularities of a given hyperplane section are mild, one can ask whether the same holds for the ambient variety. Such inversion of adjunction questions have a fairly satisfactory answer in both equal characteristic zero and p>0, and in this talk we aim to address this problem in mixed characteristic. Using the framework of perfectoid big Cohen-Macaulay algebras, we define a class of singularities satisfying adjunction and inversion of adjunction properties analogous to those for PLT singularities in complex algebraic geometry (or purely F-regular singularities in characteristic p>0). As an application, we obtain a new form of the Briancon-Skoda theorem in mixed characteristic.</div>Andreichttps://www.math.wisc.edu/wiki/index.php?title=Algebra_and_Algebraic_Geometry_Seminar_Spring_2020&diff=19030Algebra and Algebraic Geometry Seminar Spring 20202020-02-14T16:41:44Z<p>Andreic: </p>
<hr />
<div>== Spring 2020 Schedule ==<br />
<br />
{| cellpadding="8"<br />
!align="left" | date<br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s) <br />
|-<br />
|January 24<br />
|[http://www.math.ualberta.ca/~xichen// Xi Chen (Alberta)]<br />
|[[#Xi Chen|Rational Curves on K3 Surfaces]]<br />
|Michael K<br />
|-<br />
|January 31<br />
|[http://www.math.utah.edu/~letz// Janina Letz (Utah)]<br />
|[[#Janina Letz|Local to global principles for generation time over commutative rings]]<br />
|Daniel and Michael B<br />
|-<br />
|February 7<br />
|Jonathan Monta&#241;o (New Mexico State)<br />
|Asymptotic behavior of invariants of symbolic powers<br />
|Daniel<br />
|-<br />
|February 14<br />
|<br />
|<br />
| <br />
|-<br />
|February 21<br />
|Erika Ordog (Duke)<br />
|Minimal resolutions of monomial ideals<br />
|Daniel<br />
|-<br />
|February 28<br />
|<br />
|<br />
|<br />
|-<br />
|March 6<br />
|<br />
|<br />
|<br />
|-<br />
|March 13<br />
|Kevin Tucker (UIC)<br />
|TBD<br />
|Daniel<br />
|-<br />
|March 20<br />
|<br />
|<br />
|<br />
|-<br />
|March 27<br />
|[https://mcfaddin.github.io// Patrick McFaddin (Fordham)]<br />
|TBD<br />
|Michael B<br />
|-<br />
|March 30, 3:30pm '''(Monday, unusual date and time!)'''<br />
|[https://pages.uoregon.edu/honigs/ Katrina Honigs (Oregon)]<br />
|TBA<br />
|Andrei<br />
|-<br />
|April 3<br />
|<br />
| <br />
| <br />
|-<br />
|April 10<br />
|[https://sites.google.com/view/ruijie-yang/ Ruijie Yang (Stony Brook)]<br />
|TBD<br />
|Michael K<br />
|-<br />
|April 17<br />
|Remy van Dobben de Bruyn (Princeton/IAS)<br />
|TBD<br />
|Botong<br />
|-<br />
|April 24<br />
|Katrina Honigs (University of Oregon)<br />
|TBA<br />
|Andrei<br />
|-<br />
|May 1<br />
|Lazarsfeld Distinguished Lectures<br />
|<br />
|<br />
|-<br />
|May 8<br />
|<br />
|<br />
| <br />
|-<br />
|}<br />
<br />
== Abstracts ==<br />
===Xi Chen===<br />
'''Rational Curves on K3 Surfaces<br />
'''<br />
<br />
It is conjectured that there are infinitely many rational<br />
curves on every projective K3 surface. A large part of this conjecture<br />
was proved by Jun Li and Christian Liedtke, based on the<br />
characteristic p reduction method proposed by<br />
Bogomolov-Hassett-Tschinkel. They proved that there are infinitely<br />
many rational curves on every projective K3 surface of odd Picard<br />
rank. Over complex numbers, there are a few remaining cases: K3<br />
surfaces of Picard rank two excluding elliptic K3's and K3's with<br />
infinite automorphism groups and K3 surfaces with two particular<br />
Picard lattices of rank four. We have settled these leftover cases and also<br />
generalized the conjecture to the existence of curves of high genus.<br />
This is a joint work with Frank Gounelas and Christian Liedtke.<br />
<br />
===Janina Letz===<br />
'''Local to global principles for generation time over commutative rings<br />
'''<br />
<br />
Abstract: In the derived category of modules over a commutative<br />
noetherian ring a complex $G$ is said to generate a complex $X$ if the<br />
latter can be obtained from the former by taking finitely many summands<br />
and cones. The number of cones needed in this process is the generation<br />
time of $X$. In this talk I will present some local to global type<br />
results for computing this invariant, and also discuss some<br />
applications of these results.<br />
<br />
<br />
===Jonathan Montano===<br />
'''Asymptotic behavior of invariants of symbolic powers '''<br />
<br />
Abstract: The symbolic powers of an ideal is a filtration that encodes important algebraic and geometric information of the ideal and the variety it defines. Despite the importance and great results about symbolic powers, their complete structure is far from being understood. For example, we do not completely understand yet the behavior of the number of generators, regularities, and depths of these ideals. In this talk I will report on resent results in this direction in joint works with Hailong Dao and Luis Núñez-Betancourt.<br />
<br />
===Erika Ordog===<br />
'''Minimal resolutions of monomial ideals'''<br />
<br />
Abstract: The problem of finding minimal free resolutions of monomial<br />
ideals in polynomial rings has been central to commutative<br />
algebra ever since Kaplansky raised the problem in the 1960s and<br />
his student, Diana Taylor, produced the first general<br />
construction in 1966. The ultimate goal along these lines is a<br />
construction of free resolutions that is universal -- that is,<br />
valid for arbitrary monomial ideals -- canonical, combinatorial,<br />
and minimal. This talk describes a solution to the problem<br />
valid in characteristic 0 and almost all positive characteristics.<br />
<br />
<br />
===Kevin Tucker===<br />
'''An Analog of PLT Singularities in Mixed Characteristic'''<br />
<br />
Abstract: In algebraic geometry, singularities are often understood using hyperplane sections. For example, if the singularities of a given hyperplane section are mild, one can ask whether the same holds for the ambient variety. Such inversion of adjunction questions have a fairly satisfactory answer in both equal characteristic zero and p>0, and in this talk we aim to address this problem in mixed characteristic. Using the framework of perfectoid big Cohen-Macaulay algebras, we define a class of singularities satisfying adjunction and inversion of adjunction properties analogous to those for PLT singularities in complex algebraic geometry (or purely F-regular singularities in characteristic p>0). As an application, we obtain a new form of the Briancon-Skoda theorem in mixed characteristic.</div>Andreichttps://www.math.wisc.edu/wiki/index.php?title=Algebra_and_Algebraic_Geometry_Seminar_Spring_2020&diff=19029Algebra and Algebraic Geometry Seminar Spring 20202020-02-14T16:23:58Z<p>Andreic: </p>
<hr />
<div>== Spring 2020 Schedule ==<br />
<br />
{| cellpadding="8"<br />
!align="left" | date<br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s) <br />
|-<br />
|January 24<br />
|[http://www.math.ualberta.ca/~xichen// Xi Chen (Alberta)]<br />
|[[#Xi Chen|Rational Curves on K3 Surfaces]]<br />
|Michael K<br />
|-<br />
|January 31<br />
|[http://www.math.utah.edu/~letz// Janina Letz (Utah)]<br />
|[[#Janina Letz|Local to global principles for generation time over commutative rings]]<br />
|Daniel and Michael B<br />
|-<br />
|February 7<br />
|Jonathan Monta&#241;o (New Mexico State)<br />
|Asymptotic behavior of invariants of symbolic powers<br />
|Daniel<br />
|-<br />
|February 14<br />
|<br />
|<br />
| <br />
|-<br />
|February 21<br />
|Erika Ordog (Duke)<br />
|Minimal resolutions of monomial ideals<br />
|Daniel<br />
|-<br />
|February 28<br />
|<br />
|<br />
|<br />
|-<br />
|March 6<br />
|<br />
|<br />
|<br />
|-<br />
|March 13<br />
|Kevin Tucker (UIC)<br />
|TBD<br />
|Daniel<br />
|-<br />
|March 20<br />
|<br />
|<br />
|<br />
|-<br />
|March 27<br />
|[https://mcfaddin.github.io// Patrick McFaddin (Fordham)]<br />
|TBD<br />
|Michael B<br />
|-<br />
|March 30 '''(Monday, unusual date and time!)'''<br />
|[https://pages.uoregon.edu/honigs/ Katrina Honigs (Oregon)]<br />
|TBA<br />
|Andrei<br />
|-<br />
|April 3<br />
|<br />
| <br />
| <br />
|-<br />
|April 10<br />
|[https://sites.google.com/view/ruijie-yang/ Ruijie Yang (Stony Brook)]<br />
|TBD<br />
|Michael K<br />
|-<br />
|April 17<br />
|Remy van Dobben de Bruyn (Princeton/IAS)<br />
|TBD<br />
|Botong<br />
|-<br />
|April 24<br />
|Katrina Honigs (University of Oregon)<br />
|TBA<br />
|Andrei<br />
|-<br />
|May 1<br />
|Lazarsfeld Distinguished Lectures<br />
|<br />
|<br />
|-<br />
|May 8<br />
|<br />
|<br />
| <br />
|-<br />
|}<br />
<br />
== Abstracts ==<br />
===Xi Chen===<br />
'''Rational Curves on K3 Surfaces<br />
'''<br />
<br />
It is conjectured that there are infinitely many rational<br />
curves on every projective K3 surface. A large part of this conjecture<br />
was proved by Jun Li and Christian Liedtke, based on the<br />
characteristic p reduction method proposed by<br />
Bogomolov-Hassett-Tschinkel. They proved that there are infinitely<br />
many rational curves on every projective K3 surface of odd Picard<br />
rank. Over complex numbers, there are a few remaining cases: K3<br />
surfaces of Picard rank two excluding elliptic K3's and K3's with<br />
infinite automorphism groups and K3 surfaces with two particular<br />
Picard lattices of rank four. We have settled these leftover cases and also<br />
generalized the conjecture to the existence of curves of high genus.<br />
This is a joint work with Frank Gounelas and Christian Liedtke.<br />
<br />
===Janina Letz===<br />
'''Local to global principles for generation time over commutative rings<br />
'''<br />
<br />
Abstract: In the derived category of modules over a commutative<br />
noetherian ring a complex $G$ is said to generate a complex $X$ if the<br />
latter can be obtained from the former by taking finitely many summands<br />
and cones. The number of cones needed in this process is the generation<br />
time of $X$. In this talk I will present some local to global type<br />
results for computing this invariant, and also discuss some<br />
applications of these results.<br />
<br />
<br />
===Jonathan Montano===<br />
'''Asymptotic behavior of invariants of symbolic powers '''<br />
<br />
Abstract: The symbolic powers of an ideal is a filtration that encodes important algebraic and geometric information of the ideal and the variety it defines. Despite the importance and great results about symbolic powers, their complete structure is far from being understood. For example, we do not completely understand yet the behavior of the number of generators, regularities, and depths of these ideals. In this talk I will report on resent results in this direction in joint works with Hailong Dao and Luis Núñez-Betancourt.<br />
<br />
===Erika Ordog===<br />
'''Minimal resolutions of monomial ideals'''<br />
<br />
Abstract: The problem of finding minimal free resolutions of monomial<br />
ideals in polynomial rings has been central to commutative<br />
algebra ever since Kaplansky raised the problem in the 1960s and<br />
his student, Diana Taylor, produced the first general<br />
construction in 1966. The ultimate goal along these lines is a<br />
construction of free resolutions that is universal -- that is,<br />
valid for arbitrary monomial ideals -- canonical, combinatorial,<br />
and minimal. This talk describes a solution to the problem<br />
valid in characteristic 0 and almost all positive characteristics.<br />
<br />
<br />
===Kevin Tucker===<br />
'''An Analog of PLT Singularities in Mixed Characteristic'''<br />
<br />
Abstract: In algebraic geometry, singularities are often understood using hyperplane sections. For example, if the singularities of a given hyperplane section are mild, one can ask whether the same holds for the ambient variety. Such inversion of adjunction questions have a fairly satisfactory answer in both equal characteristic zero and p>0, and in this talk we aim to address this problem in mixed characteristic. Using the framework of perfectoid big Cohen-Macaulay algebras, we define a class of singularities satisfying adjunction and inversion of adjunction properties analogous to those for PLT singularities in complex algebraic geometry (or purely F-regular singularities in characteristic p>0). As an application, we obtain a new form of the Briancon-Skoda theorem in mixed characteristic.</div>Andreichttps://www.math.wisc.edu/wiki/index.php?title=Math_567_--_Elementary_Number_Theory&diff=19008Math 567 -- Elementary Number Theory2020-02-12T19:05:45Z<p>Andreic: </p>
<hr />
<div>'''MATH 567<br />
<br />
Elementary Number Theory'''<br />
<br />
MWF 1:20-2:10, Van Vleck B119<br />
<br />
'''Professor:''' [http://www.math.wisc.edu/~andreic/ Andrei Caldararu] (andreic@math.wisc.edu)<br />
''Office Hours:'' Wednesdays 2:30-3:30, Van Vleck 605. <br />
<br />
'''Grader:''' TBA<br />
''Office Hours:'' TBA.<br />
<br />
Math 567 is a course in elementary number theory, aimed at undergraduates majoring in math or other quantitative disciplines. A general familiarity with abstract algebra at the level of Math 541 will be assumed, but students who haven't taken 541 are welcome to attend if they're willing to play a little catchup. We will be using William Stein's new (and cheap) textbook [http://www.amazon.com/Elementary-Number-Theory-Computational-Undergraduate/dp/0387855246 Elementary Number Theory: Primes, Congruences, and Secrets], which emphasizes computational approaches to the subject. If you don't need a physical copy of the book, [http://www.williamstein.org/ent/ it is available as a free legal .pdf.] We will be using the (free, public-domain) mathematical software [http://www.sagemath.org/ SAGE], developed largely by Stein, as an integral component of our coursework. There is a [http://doc.sagemath.org/pdf/en/tutorial/SageTutorial.pdf useful online tutorial.] You can download SAGE to your own computer or [http://www.sagenb.org use it online].<br />
<br />
Topics include some subset of, but are not limited to: Divisibility, the Euclidean algorithm and the GCD, linear Diophantine equations, prime numbers and uniqueness of factorization. Congruences, Chinese remainder theorem, Fermat's "little" theorem, Wilson's theorem, Euler's theorem and totient function, the RSA cryptosystem, Rabin's encryption scheme, Diffie-Hellman key exchange protocol. Number-theoretic functions, multiplicative functions, Möbius inversion. Primitive roots and indices. Quadratic reciprocity and the Legendre symbol. Perfect numbers, Mersenne primes, Fermat primes. Pythagorean triples, Special cases of Fermat's "last" theorem. Fibonacci numbers. Continued fractions. Distribution of primes, discussion of prime number theorem. Primality testing and factoring algorithms.<br />
<br />
'''Course Policies:''' Homework will be due on Fridays. It can be turned in late only with ''advance'' permission from your grader. It is acceptable to use calculators and computers on homework (indeed, some of it will require a computer) but calculators are not allowed during exams. You are encouraged to work together on homework, but writeups must be done individually.<br />
<br />
Many of the problems in this course will ask you to prove things. I expect proofs to be written in English sentences; the proofs in Stein's book are a good model for the level of verbosity I am looking for. <br />
<br />
'''Grading:''' The grade in Math 567 will be composed of 50% homework, 25% midterm, 25% final. The midterm will be on March 11, in class. The final exam will be on 5/8/2020, 7:45AM-9:45AM in room TBA.<br />
<br />
'''Syllabus:''' <br />
(This may change as we see what pace works well for the course. All section numbers refer to Stein's book.)<br />
<br />
* Jan 22-31: Prime numbers, prime factorizations, Euclidean algorithm and GCD (1.1-1.2)<br />
* Feb 3-7: The integers mod n, Euler's theorem, the phi function (2.1-2.2)<br />
* Feb 10-14: Modular exponentiation, primality testing, and primitive roots (2.4-2.5)<br />
* Feb 17-21: Public-key cryptography and RSA (3.1-3.4)<br />
* Feb 24-28: Rabin's algorithm (not in the book); algebraic numbers <br />
* Mar 2-6: Quadratic reciprocity (4.1-4.4)<br />
* Mar 9, Mar 13: Finite and infinite continued fractions (5.1-5.3)<br />
* Mar 11: ''Midterm exam''<br />
* Mar 23-30: Continued fractions and diophantine approximation (5.4-5.5)<br />
* Apr 1-3: Diophantine equations I: Pell's equation and Lagrange's theorem<br />
* Apr 6-10: More diophantine equations, elliptic curves (6.1)<br />
* Apr 13-17: Applications of elliptic curves (6.2-6.3)<br />
* Apr 20-24: More applications of elliptic curves (6.4)<br />
* Apr 27-May 1: advanced topic TBD: maybe additional discussion of cryptographic techniques?<br />
<br />
'''Homework:'''<br />
Homework is due at the beginning of class on the specified Friday. Typing your homework is not a requirement, but if you don't already know LaTeX I highly recommend that you learn it and use it to typeset your homework. I will sometimes assign extra problems, which I will e-mail to the class list and include here.<br />
<br />
* '''Jan 31''': 1.1, 1.3, 1.5, 1.7 (use SAGE), 1.8, 1.14.<br />
Problem A: Use SAGE to compute the number of x in [1..N] such that x^2 + 1 is prime, for N = 100, N = 1000, and N = 10000. Let f(N) be the number of such x. <br />
<br />
a) Can you formulate a conjecture about the relationship between f(N) and N/log N? <br />
<br />
b) What if x^2 + 1 is replaced with x^2 + 2? Can you explain why x^2 + 2 appears less likely to be prime? (Hint: consider x mod 3.)<br />
<br />
c) Prove that f(N) is at most (1/2)N+1. (Hint: consider x mod 2.) <br />
<br />
d) Give as good an upper bound as you can for f(N).<br />
<br />
Note that, despite the evident regularities you'll observe in this problem, we do not even know whether there are infinitely many primes of the form x^2 + 1! You would become very famous if you proved this.<br />
<br />
* '''Feb 7''': 2.6 (the formulation of numerical evidence should be done by Sage if you've got Sage working, and by calculator if not; you can use an online tool like [http://primes.utm.edu/curios/includes/primetest.php this] to test whether a number is prime.) 2.8,2.9,2.11,2.12,2.14,2.19<br />
<br />
* '''Feb 17''': 2.15, 2.18, 2.23, 2.26, 2.30.<br />
<br />
Problem A: Prove that if n=pq, with p,q prime, then n is not a Carmichael number.<br />
<br />
<!--<br />
* '''Oct 12''': Book problems: 3.4, 3.5, 3.6<br />
<br />
Problem A. Prove that there are infinitely many primes p such that 2 is '''not''' a primitive root in Z/pZ. We break this up into steps.<br />
<br />
Problem A.1. Prove that, if x is an element of Z/nZ, then x^2 is not a primitive root.<br />
<br />
Problem A.2. Prove that there are infinitely many primes p such that 2 is a square in Z/pZ. Hint: suppose there are only finitely many such primes p_1, .. p_r, and define N = (p_1 .. p_r)^2 - 2. Where can you go from here...?<br />
<br />
Problem A.3. Give a list of five primes p such that 2 is not a primitive root in Z/pZ (you can use the method of this proof or any other.)<br />
<br />
Problem B. Prove that 24 is the largest integer n such that every element of (Z/nZ)^* is a root of x^2-1.<br />
<br />
* '''Oct 19''': Problem A. Using p = 23 and q=31, show how to encrypt the message x=240 with Rabin's algorithm. Find all possible decryptions of your encrypted message.<br />
<br />
Problem B. Give a prime factorization of the Gaussian integer 7+9i.<br />
<br />
Problem C. Read the notes from [http://www.math.ucsd.edu/~jverstra/Gaussian1.pdf here]. Note that Z[i] satisfies a ''reduction theorem'': if n and d are Gaussian integers, then there exists integers q and r such that n = qd + r and Norm(r) < Norm(d). But (by contrast with the case of Z) this d may not be unique. In some contexts it is better to be able to choose r uniquely, even if this means letting r have norm greater than Norm(d). <br />
<br />
C.1. When d = 1+2i, show that, for each n in Z[i], there is a '''unique''' pair (q,r) in Z[i] such that n = qd+r and r is contained in the set {0,1,2,3,4}. For instance, i can be written as i(1+2i) + 2, so we say i reduces to 2 mod (1+2i).<br />
(Hint: Suppose q(1+2i)+r = q'(1+2i) + r'. What can we say about (r-r'), and why is this incompatible with both r and r' being in {0,1,2,3,4}?<br />
<br />
C.2. Show that if n is an integer in Z, the reduction of n mod (1+2i) is equal to its reduction mod 5.<br />
<br />
Problem D. Let's try to figure out how to define "phi(d)" for a Gaussian integer d. Suppose S is a set of Gaussian integers such that every n in Z[i] can be written uniquely as qd+r, with q in Z[i] and r in S. (So for instance when d=1+2i, we showed in problem B that we can take S to be {0...4}. It would also be OK to take S to be {1..5} or {0,i,2i,1+i,1+2i}. In fact, it turns out that S has to be a set of size Norm(d) (I might or might not prove this in class; if not, feel free just to accept it.)<br />
<br />
Now define phi(d) to be the number of elements s of S such that s and d are coprime.<br />
<br />
D.1. Compute phi(1+2i) and phi(3).<br />
<br />
D.2. Prove that the value of phi(d) does not depend on the choice of S.<br />
<br />
D.3. Prove that every n in Z[i] which is coprime to 3 satisfies n^phi(3) = 1 mod 3; that is, Euler's theorem holds in this case. (You can prove this by direct computation; of course, if you want, you are welcome to prove that Euler's theorem holds for Z[i] in general, adapting the proof in Stein's book or the one we gave in class.)<br />
<br />
*'''Nov 2:''' 4.1 from the book.<br />
<br />
Problem A. Express 50005 as the sum of two squares.<br />
<br />
In the next two problems we denote by r(n) the number of ways to express n as the sum of two squares (i.e. the number of pairs (a,b) such that a^2 + b^2 = n.) For instance, r(5) = 8 (as shown on the midterm.)<br />
<br />
Problem B. Prove that, for any N, there exists an integer n such that r(n) > N. (I.E., the function r(n) is ''unbounded.''<br />
<br />
Problem C. If you like Sage, write a short program in Sage to compute r(n) and compute the average of r(n) as n ranges from 1 to 1000. Whether or not you like Sage, make a guess as to how this average would behave if you replaced "1000" by a larger and larger number. (Feel free to ask the Sage-lovers what answer they got in the optional first part of the question.) Can you prove this guess is correct?<br />
<br />
Problem D. We saw in class that the ring Z[sqrt(-5)] doesn't have unique factorization; 6 can be factored as 2*3 or (1+sqrt(-5))(1-sqrt(-5)). In this problem, we will prove that Z[sqrt(-d)] fails to have unique factorization for EVERY odd d >= 5. (Actually it's true for all d >= 5 but to make the proof manageable we'll restrict to the odd case.]<br />
<br />
D.1. Show that (1+sqrt(-d)), (1-sqrt(-d)) and 2 are irreducible in Z[sqrt(-d)].<br />
<br />
D.2. Now give an element of Z[sqrt(-d)] that has two distinct factorizations into irreducibles. (Hint: imitate the example we used for Z[sqrt(-5)].)<br />
''Remark:'' The rings Z[sqrt(d)], where d is positive, are quite different -- here we believe that there are infinitely many with unique factorization, though this conjecture has remained unproved for many decades!<br />
<br />
*'''November 16'''<br />
<br />
Book problems: 5.3,5.4,5.5<br />
<br />
Problem A: We discussed in class the problem of studying which positive integers are the sum of two squares. In this problem we prove that every element n of (Z/pZ) is the sum of two squares. We argue as follows. Let S be the set of squares in (Z/pZ), and let T be the set of (Z/pZ) consisting of all elements of the form n - x^2, for some x in (Z/pZ).<br />
<br />
A.1. What is the size of S and of T? Use this to show that S and T are not disjoint.<br />
<br />
A.2. Given that S and T are not disjoint, prove that n is the sum of two squares.<br />
<br />
Problem B. Using the continued fraction expansion, find a solution to the Pell equation x^2 - 13 y^2 = 1.<br />
<br />
Problem C. Show that the modified Pell equation x^2 - 7y^2 = -1 has no solutions in integers x,y. (Hint: reduce the equation modulo a suitably chosen prime.)<br />
<br />
Problem D. Stuffy Stirnweiss finished the 1945 season with a batting average of .3085443. Using continued fractions, guess how many at-bats he had. Tony Cuccinello had a batting average of .3084577. Given that he had more than 200 and fewer than 600 at-bats, can you estimate the number of at-bats he had?<br />
<br />
'''December 7'''<br />
<br />
Problem A. When p is a prime congruent to 3 mod 4, prove that ((p-1)/2)! is either 1 or -1 in (Z/pZ). '''OPTIONAL:''' Use sage to compute whether this factorial is 1 or -1 for many primes p. Is there any pattern? Does it seem to be 1 half the time and -1 half the time? Note that I have no idea what the answer to this question is.<br />
<br />
Problem B. Using the continued fraction expansion, find a solution to the Pell equation x^2 - 13 y^2 = 1.<br />
<br />
Problem C. Show that the modified Pell equation x^2 - 7y^2 = -1 has no solutions in integers x,y. (Hint: reduce the equation modulo a suitably chosen prime.)<br />
<br />
Problem D.1. Pick two values of a in F_11 = Z/11Z (a not equal to 3), such that the equation y^2 = x^3+ax+1 defines an elliptic curve (i.e., it is smooth). For each such a, determine the number of points #E(F_11) and check that it falls inside the interval described in class. <br />
<br />
Problem D.2. The point P = (0,1) lies on each of these curves. For a =3 (the curve discussed in class) determine the order of P in the elliptic curve group, that is, find the smallest positive integer n such that nP = (infinity) -- the identity element in the group law.<br />
<br />
<!-- '''November 19'''<br />
<br />
Problem A. Using the method discussed in class (which is also the method of problem 4.6 in Stein, which in retrospect I think was too hard to assign with no preparation) find a nontrivial solution with y > 0 to the equation x^2 + 11y^2 = z^2. <br />
Problem B. Let f(X) be the number of solutions of A^2 + B^2 = C^3 such that A^2, B^2, and C^3 are all at most X. Use the heuristic described in class to explain why one might expect f(X) to grow more or less like X^{1/3}.<br />
<br />
Problem C. Show that if A+Bi is the cube of a Gaussian integer, then A^2 + B^2 is a perfect cube.<br />
<br />
Do ONE of problem D1 and D2; D1 is for people who like Sage, D2 is for people who like proving things.<br />
<br />
D1. Use Sage to compute f(X) for X = 1000, 10000, 100000. Does the answer look consistent with the heuristic prediction you made in B? Does f(X)/X^{1/3} appear to be approaching a limit?<br />
D2. We will use problem C to give a lower bound for f(X). Use this to show that there are at least X^{1/3} Gaussian integers with norm at most X that are perfect cubes. From here, show that f(X) > X^{1/3}.<br />
<br />
OPTIONAL: Show that the converse of problem C also holds -- A+Bi has norm a perfect cube if and only if A+Bi is a perfect cube. (You will need unique factorization of Gaussian integers.) Using this fact, prove that f(X) is bounded above by C X^{1/3} for some constant C.<br />
<br />
'''December 8 (note nonstandard due date)'''<br />
<br />
Problem A: Hyperbolas, ellipses, and "magic slopes"<br />
<br />
In the usual analytic geometry, both hyperbolas and ellipses are given by equations of the form<br />
<br />
ax^2 + bxy + cy^2 = d (*)<br />
<br />
<br />
(we always assume d is nonzero.)<br />
<br />
How can we tell whether such an equation describes an ellipse or a<br />
hyperbola? Well, in the geometric setting we all know and love (i.e. in<br />
R^2) a hyperbola has asymptotes and an ellipse does not. What is an <br />
asymptote? You could say it's "a line which doesn't intersect the curve", <br />
but ellipses have such lines too. You might want to say "a line which <br />
doesn't intersect the curve but comes closer and closer to the curve," but <br />
the problem here is that this a) is somewhat imprecise, and b) relies on a <br />
notion of "closer" that is not going to be very clear in Z/pZ, which is <br />
our ultimate goal!<br />
<br />
<br />
So let me state it a different way: let's say the asymptotes of a <br />
hyperbola have slopes m_1 and m_2. These are "magic slopes" in the <br />
following sense: ANY line of slope m_1 (and ditto for m_2) intersects the <br />
hyperbola in at most one point. An ellipse doesn't have "magic slopes" -- <br />
you can convince yourself by drawing pictures that, for any slope m, you <br />
can find a line of slope m striking the ellipse twice.<br />
<br />
<br />
[REMARK: For technical reasons we're going to ignore hyperbolae with a <br />
vertical asymptote, since vertical lines don't exactly have slope.]<br />
<br />
<br />
Now you might know the criterion that, over the real numbers.<br />
<br />
<br />
ax^2 + bxy + cy^2 = d<br />
<br />
<br />
is a hyperbola if b^2 - 4ac > 0 and an ellipse if b^2 - 4ac < 0. (For <br />
instance, x^2 - y^2 = 1 is a hyperbola with magic slopes 1 and -1, and x^2 <br />
+ y^2 = 1 is an ellipse.) <br />
This criterion does not make sense in Z/pZ, where there is no notion of <br />
"greater" or "less."<br />
<br />
<br />
QUESTION A.1.<br />
<br />
For which primes p is it the case that <br />
<br />
x^2 + y^2 = 1<br />
<br />
has magic slopes in Z/pZ? <br />
<br />
QUESTION A.2. For which primes p is it the case that<br />
<br />
<br />
x^2 - y^2 = 1<br />
<br />
<br />
has magic slopes in Z/pZ?<br />
<br />
<br />
QUESTION B: Back in the very first week of this course we studied the set of integers x such that x^2 + 1 is prime. Based on the heuristics we discussed in class, about how many x between 1 and N would you expect to satisfy "x^2 + 1 is prime?"<br />
<br />
<br />
QUESTION C: Let z be a complex number. Then the set of all complex numbers of the form a + bz, with a and b integers, forms a lattice in the complex plane (thought of as R^2.) What is the covolume of this lattice, in terms of z?<br />
<br />
<br />
QUESTION D: Let z be a complex number. Using Minkowski's theorem and the result of question C, show that there exist integers a,b, not both zero, such that Norm(a+bz) is at most (4/pi) Im(z). (Hint: let Omega be the set of all complex numbers of norm at most (4/pi) Im(z).)<br />
<br />
<br />
OPTIONAL EXTRA: Can you give an exact formula for the minimal norm of any nonzero complex number of the form a+bz?<br />
<br />
'''December 12''' Book problems 6.1, 6.2, 6.5, 6.10. For extra credit attempt to do problem 6.9.<br />
<br />
--></div>Andreichttps://www.math.wisc.edu/wiki/index.php?title=Math_567_--_Elementary_Number_Theory&diff=19007Math 567 -- Elementary Number Theory2020-02-12T19:05:17Z<p>Andreic: </p>
<hr />
<div>'''MATH 567<br />
<br />
Elementary Number Theory'''<br />
<br />
MWF 1:20-2:10, Van Vleck B119<br />
<br />
'''Professor:''' [http://www.math.wisc.edu/~andreic/ Andrei Caldararu] (andreic@math.wisc.edu)<br />
''Office Hours:'' Wednesdays 2:30-3:30, Van Vleck 605. <br />
<br />
'''Grader:''' TBA<br />
''Office Hours:'' TBA.<br />
<br />
Math 567 is a course in elementary number theory, aimed at undergraduates majoring in math or other quantitative disciplines. A general familiarity with abstract algebra at the level of Math 541 will be assumed, but students who haven't taken 541 are welcome to attend if they're willing to play a little catchup. We will be using William Stein's new (and cheap) textbook [http://www.amazon.com/Elementary-Number-Theory-Computational-Undergraduate/dp/0387855246 Elementary Number Theory: Primes, Congruences, and Secrets], which emphasizes computational approaches to the subject. If you don't need a physical copy of the book, [http://www.williamstein.org/ent/ it is available as a free legal .pdf.] We will be using the (free, public-domain) mathematical software [http://www.sagemath.org/ SAGE], developed largely by Stein, as an integral component of our coursework. There is a [http://doc.sagemath.org/pdf/en/tutorial/SageTutorial.pdf useful online tutorial.] You can download SAGE to your own computer or [http://www.sagenb.org use it online].<br />
<br />
Topics include some subset of, but are not limited to: Divisibility, the Euclidean algorithm and the GCD, linear Diophantine equations, prime numbers and uniqueness of factorization. Congruences, Chinese remainder theorem, Fermat's "little" theorem, Wilson's theorem, Euler's theorem and totient function, the RSA cryptosystem, Rabin's encryption scheme, Diffie-Hellman key exchange protocol. Number-theoretic functions, multiplicative functions, Möbius inversion. Primitive roots and indices. Quadratic reciprocity and the Legendre symbol. Perfect numbers, Mersenne primes, Fermat primes. Pythagorean triples, Special cases of Fermat's "last" theorem. Fibonacci numbers. Continued fractions. Distribution of primes, discussion of prime number theorem. Primality testing and factoring algorithms.<br />
<br />
'''Course Policies:''' Homework will be due on Fridays. It can be turned in late only with ''advance'' permission from your grader. It is acceptable to use calculators and computers on homework (indeed, some of it will require a computer) but calculators are not allowed during exams. You are encouraged to work together on homework, but writeups must be done individually.<br />
<br />
Many of the problems in this course will ask you to prove things. I expect proofs to be written in English sentences; the proofs in Stein's book are a good model for the level of verbosity I am looking for. <br />
<br />
'''Grading:''' The grade in Math 567 will be composed of 50% homework, 25% midterm, 25% final. The midterm will be on March 11, in class. The final exam will be on 5/8/2020, 7:45AM-9:45AM in room TBA.<br />
<br />
'''Syllabus:''' <br />
(This may change as we see what pace works well for the course. All section numbers refer to Stein's book.)<br />
<br />
* Jan 22-31: Prime numbers, prime factorizations, Euclidean algorithm and GCD (1.1-1.2)<br />
* Feb 3-7: The integers mod n, Euler's theorem, the phi function (2.1-2.2)<br />
* Feb 10-14: Modular exponentiation, primality testing, and primitive roots (2.4-2.5)<br />
* Feb 17-21: Public-key cryptography and RSA (3.1-3.4)<br />
* Feb 24-28: Rabin's algorithm (not in the book); algebraic numbers <br />
* Mar 2-6: Quadratic reciprocity (4.1-4.4)<br />
* Mar 9, Mar 13: Finite and infinite continued fractions (5.1-5.3)<br />
* Mar 11: ''Midterm exam''<br />
* Mar 23-30: Continued fractions and diophantine approximation (5.4-5.5)<br />
* Apr 1-3: Diophantine equations I: Pell's equation and Lagrange's theorem<br />
* Apr 6-10: More diophantine equations, elliptic curves (6.1)<br />
* Apr 13-17: Applications of elliptic curves (6.2-6.3)<br />
* Apr 20-24: More applications of elliptic curves (6.4)<br />
* Apr 27-May 1: advanced topic TBD: maybe additional discussion of cryptographic techniques?<br />
<br />
'''Homework:'''<br />
Homework is due at the beginning of class on the specified Friday. Typing your homework is not a requirement, but if you don't already know LaTeX I highly recommend that you learn it and use it to typeset your homework. I will sometimes assign extra problems, which I will e-mail to the class list and include here.<br />
<br />
* '''Jan 31''': 1.1, 1.3, 1.5, 1.7 (use SAGE), 1.8, 1.14.<br />
Problem A: Use SAGE to compute the number of x in [1..N] such that x^2 + 1 is prime, for N = 100, N = 1000, and N = 10000. Let f(N) be the number of such x. <br />
<br />
a) Can you formulate a conjecture about the relationship between f(N) and N/log N? <br />
<br />
b) What if x^2 + 1 is replaced with x^2 + 2? Can you explain why x^2 + 2 appears less likely to be prime? (Hint: consider x mod 3.)<br />
<br />
c) Prove that f(N) is at most (1/2)N+1. (Hint: consider x mod 2.) <br />
<br />
d) Give as good an upper bound as you can for f(N).<br />
<br />
Note that, despite the evident regularities you'll observe in this problem, we do not even know whether there are infinitely many primes of the form x^2 + 1! You would become very famous if you proved this.<br />
<br />
* Feb 7: 2.6 (the formulation of numerical evidence should be done by Sage if you've got Sage working, and by calculator if not; you can use an online tool like [http://primes.utm.edu/curios/includes/primetest.php this] to test whether a number is prime.) 2.8,2.9,2.11,2.12,2.14,2.19<br />
<br />
* Feb 17: 2.15, 2.18, 2.23, 2.26, 2.30.<br />
<br />
Problem A: Prove that if n=pq, with p,q prime, then n is not a Carmichael number.<br />
<br />
<!--<br />
* '''Oct 12''': Book problems: 3.4, 3.5, 3.6<br />
<br />
Problem A. Prove that there are infinitely many primes p such that 2 is '''not''' a primitive root in Z/pZ. We break this up into steps.<br />
<br />
Problem A.1. Prove that, if x is an element of Z/nZ, then x^2 is not a primitive root.<br />
<br />
Problem A.2. Prove that there are infinitely many primes p such that 2 is a square in Z/pZ. Hint: suppose there are only finitely many such primes p_1, .. p_r, and define N = (p_1 .. p_r)^2 - 2. Where can you go from here...?<br />
<br />
Problem A.3. Give a list of five primes p such that 2 is not a primitive root in Z/pZ (you can use the method of this proof or any other.)<br />
<br />
Problem B. Prove that 24 is the largest integer n such that every element of (Z/nZ)^* is a root of x^2-1.<br />
<br />
* '''Oct 19''': Problem A. Using p = 23 and q=31, show how to encrypt the message x=240 with Rabin's algorithm. Find all possible decryptions of your encrypted message.<br />
<br />
Problem B. Give a prime factorization of the Gaussian integer 7+9i.<br />
<br />
Problem C. Read the notes from [http://www.math.ucsd.edu/~jverstra/Gaussian1.pdf here]. Note that Z[i] satisfies a ''reduction theorem'': if n and d are Gaussian integers, then there exists integers q and r such that n = qd + r and Norm(r) < Norm(d). But (by contrast with the case of Z) this d may not be unique. In some contexts it is better to be able to choose r uniquely, even if this means letting r have norm greater than Norm(d). <br />
<br />
C.1. When d = 1+2i, show that, for each n in Z[i], there is a '''unique''' pair (q,r) in Z[i] such that n = qd+r and r is contained in the set {0,1,2,3,4}. For instance, i can be written as i(1+2i) + 2, so we say i reduces to 2 mod (1+2i).<br />
(Hint: Suppose q(1+2i)+r = q'(1+2i) + r'. What can we say about (r-r'), and why is this incompatible with both r and r' being in {0,1,2,3,4}?<br />
<br />
C.2. Show that if n is an integer in Z, the reduction of n mod (1+2i) is equal to its reduction mod 5.<br />
<br />
Problem D. Let's try to figure out how to define "phi(d)" for a Gaussian integer d. Suppose S is a set of Gaussian integers such that every n in Z[i] can be written uniquely as qd+r, with q in Z[i] and r in S. (So for instance when d=1+2i, we showed in problem B that we can take S to be {0...4}. It would also be OK to take S to be {1..5} or {0,i,2i,1+i,1+2i}. In fact, it turns out that S has to be a set of size Norm(d) (I might or might not prove this in class; if not, feel free just to accept it.)<br />
<br />
Now define phi(d) to be the number of elements s of S such that s and d are coprime.<br />
<br />
D.1. Compute phi(1+2i) and phi(3).<br />
<br />
D.2. Prove that the value of phi(d) does not depend on the choice of S.<br />
<br />
D.3. Prove that every n in Z[i] which is coprime to 3 satisfies n^phi(3) = 1 mod 3; that is, Euler's theorem holds in this case. (You can prove this by direct computation; of course, if you want, you are welcome to prove that Euler's theorem holds for Z[i] in general, adapting the proof in Stein's book or the one we gave in class.)<br />
<br />
*'''Nov 2:''' 4.1 from the book.<br />
<br />
Problem A. Express 50005 as the sum of two squares.<br />
<br />
In the next two problems we denote by r(n) the number of ways to express n as the sum of two squares (i.e. the number of pairs (a,b) such that a^2 + b^2 = n.) For instance, r(5) = 8 (as shown on the midterm.)<br />
<br />
Problem B. Prove that, for any N, there exists an integer n such that r(n) > N. (I.E., the function r(n) is ''unbounded.''<br />
<br />
Problem C. If you like Sage, write a short program in Sage to compute r(n) and compute the average of r(n) as n ranges from 1 to 1000. Whether or not you like Sage, make a guess as to how this average would behave if you replaced "1000" by a larger and larger number. (Feel free to ask the Sage-lovers what answer they got in the optional first part of the question.) Can you prove this guess is correct?<br />
<br />
Problem D. We saw in class that the ring Z[sqrt(-5)] doesn't have unique factorization; 6 can be factored as 2*3 or (1+sqrt(-5))(1-sqrt(-5)). In this problem, we will prove that Z[sqrt(-d)] fails to have unique factorization for EVERY odd d >= 5. (Actually it's true for all d >= 5 but to make the proof manageable we'll restrict to the odd case.]<br />
<br />
D.1. Show that (1+sqrt(-d)), (1-sqrt(-d)) and 2 are irreducible in Z[sqrt(-d)].<br />
<br />
D.2. Now give an element of Z[sqrt(-d)] that has two distinct factorizations into irreducibles. (Hint: imitate the example we used for Z[sqrt(-5)].)<br />
''Remark:'' The rings Z[sqrt(d)], where d is positive, are quite different -- here we believe that there are infinitely many with unique factorization, though this conjecture has remained unproved for many decades!<br />
<br />
*'''November 16'''<br />
<br />
Book problems: 5.3,5.4,5.5<br />
<br />
Problem A: We discussed in class the problem of studying which positive integers are the sum of two squares. In this problem we prove that every element n of (Z/pZ) is the sum of two squares. We argue as follows. Let S be the set of squares in (Z/pZ), and let T be the set of (Z/pZ) consisting of all elements of the form n - x^2, for some x in (Z/pZ).<br />
<br />
A.1. What is the size of S and of T? Use this to show that S and T are not disjoint.<br />
<br />
A.2. Given that S and T are not disjoint, prove that n is the sum of two squares.<br />
<br />
Problem B. Using the continued fraction expansion, find a solution to the Pell equation x^2 - 13 y^2 = 1.<br />
<br />
Problem C. Show that the modified Pell equation x^2 - 7y^2 = -1 has no solutions in integers x,y. (Hint: reduce the equation modulo a suitably chosen prime.)<br />
<br />
Problem D. Stuffy Stirnweiss finished the 1945 season with a batting average of .3085443. Using continued fractions, guess how many at-bats he had. Tony Cuccinello had a batting average of .3084577. Given that he had more than 200 and fewer than 600 at-bats, can you estimate the number of at-bats he had?<br />
<br />
'''December 7'''<br />
<br />
Problem A. When p is a prime congruent to 3 mod 4, prove that ((p-1)/2)! is either 1 or -1 in (Z/pZ). '''OPTIONAL:''' Use sage to compute whether this factorial is 1 or -1 for many primes p. Is there any pattern? Does it seem to be 1 half the time and -1 half the time? Note that I have no idea what the answer to this question is.<br />
<br />
Problem B. Using the continued fraction expansion, find a solution to the Pell equation x^2 - 13 y^2 = 1.<br />
<br />
Problem C. Show that the modified Pell equation x^2 - 7y^2 = -1 has no solutions in integers x,y. (Hint: reduce the equation modulo a suitably chosen prime.)<br />
<br />
Problem D.1. Pick two values of a in F_11 = Z/11Z (a not equal to 3), such that the equation y^2 = x^3+ax+1 defines an elliptic curve (i.e., it is smooth). For each such a, determine the number of points #E(F_11) and check that it falls inside the interval described in class. <br />
<br />
Problem D.2. The point P = (0,1) lies on each of these curves. For a =3 (the curve discussed in class) determine the order of P in the elliptic curve group, that is, find the smallest positive integer n such that nP = (infinity) -- the identity element in the group law.<br />
<br />
<!-- '''November 19'''<br />
<br />
Problem A. Using the method discussed in class (which is also the method of problem 4.6 in Stein, which in retrospect I think was too hard to assign with no preparation) find a nontrivial solution with y > 0 to the equation x^2 + 11y^2 = z^2. <br />
Problem B. Let f(X) be the number of solutions of A^2 + B^2 = C^3 such that A^2, B^2, and C^3 are all at most X. Use the heuristic described in class to explain why one might expect f(X) to grow more or less like X^{1/3}.<br />
<br />
Problem C. Show that if A+Bi is the cube of a Gaussian integer, then A^2 + B^2 is a perfect cube.<br />
<br />
Do ONE of problem D1 and D2; D1 is for people who like Sage, D2 is for people who like proving things.<br />
<br />
D1. Use Sage to compute f(X) for X = 1000, 10000, 100000. Does the answer look consistent with the heuristic prediction you made in B? Does f(X)/X^{1/3} appear to be approaching a limit?<br />
D2. We will use problem C to give a lower bound for f(X). Use this to show that there are at least X^{1/3} Gaussian integers with norm at most X that are perfect cubes. From here, show that f(X) > X^{1/3}.<br />
<br />
OPTIONAL: Show that the converse of problem C also holds -- A+Bi has norm a perfect cube if and only if A+Bi is a perfect cube. (You will need unique factorization of Gaussian integers.) Using this fact, prove that f(X) is bounded above by C X^{1/3} for some constant C.<br />
<br />
'''December 8 (note nonstandard due date)'''<br />
<br />
Problem A: Hyperbolas, ellipses, and "magic slopes"<br />
<br />
In the usual analytic geometry, both hyperbolas and ellipses are given by equations of the form<br />
<br />
ax^2 + bxy + cy^2 = d (*)<br />
<br />
<br />
(we always assume d is nonzero.)<br />
<br />
How can we tell whether such an equation describes an ellipse or a<br />
hyperbola? Well, in the geometric setting we all know and love (i.e. in<br />
R^2) a hyperbola has asymptotes and an ellipse does not. What is an <br />
asymptote? You could say it's "a line which doesn't intersect the curve", <br />
but ellipses have such lines too. You might want to say "a line which <br />
doesn't intersect the curve but comes closer and closer to the curve," but <br />
the problem here is that this a) is somewhat imprecise, and b) relies on a <br />
notion of "closer" that is not going to be very clear in Z/pZ, which is <br />
our ultimate goal!<br />
<br />
<br />
So let me state it a different way: let's say the asymptotes of a <br />
hyperbola have slopes m_1 and m_2. These are "magic slopes" in the <br />
following sense: ANY line of slope m_1 (and ditto for m_2) intersects the <br />
hyperbola in at most one point. An ellipse doesn't have "magic slopes" -- <br />
you can convince yourself by drawing pictures that, for any slope m, you <br />
can find a line of slope m striking the ellipse twice.<br />
<br />
<br />
[REMARK: For technical reasons we're going to ignore hyperbolae with a <br />
vertical asymptote, since vertical lines don't exactly have slope.]<br />
<br />
<br />
Now you might know the criterion that, over the real numbers.<br />
<br />
<br />
ax^2 + bxy + cy^2 = d<br />
<br />
<br />
is a hyperbola if b^2 - 4ac > 0 and an ellipse if b^2 - 4ac < 0. (For <br />
instance, x^2 - y^2 = 1 is a hyperbola with magic slopes 1 and -1, and x^2 <br />
+ y^2 = 1 is an ellipse.) <br />
This criterion does not make sense in Z/pZ, where there is no notion of <br />
"greater" or "less."<br />
<br />
<br />
QUESTION A.1.<br />
<br />
For which primes p is it the case that <br />
<br />
x^2 + y^2 = 1<br />
<br />
has magic slopes in Z/pZ? <br />
<br />
QUESTION A.2. For which primes p is it the case that<br />
<br />
<br />
x^2 - y^2 = 1<br />
<br />
<br />
has magic slopes in Z/pZ?<br />
<br />
<br />
QUESTION B: Back in the very first week of this course we studied the set of integers x such that x^2 + 1 is prime. Based on the heuristics we discussed in class, about how many x between 1 and N would you expect to satisfy "x^2 + 1 is prime?"<br />
<br />
<br />
QUESTION C: Let z be a complex number. Then the set of all complex numbers of the form a + bz, with a and b integers, forms a lattice in the complex plane (thought of as R^2.) What is the covolume of this lattice, in terms of z?<br />
<br />
<br />
QUESTION D: Let z be a complex number. Using Minkowski's theorem and the result of question C, show that there exist integers a,b, not both zero, such that Norm(a+bz) is at most (4/pi) Im(z). (Hint: let Omega be the set of all complex numbers of norm at most (4/pi) Im(z).)<br />
<br />
<br />
OPTIONAL EXTRA: Can you give an exact formula for the minimal norm of any nonzero complex number of the form a+bz?<br />
<br />
'''December 12''' Book problems 6.1, 6.2, 6.5, 6.10. For extra credit attempt to do problem 6.9.<br />
<br />
--></div>Andreichttps://www.math.wisc.edu/wiki/index.php?title=Math_567_--_Elementary_Number_Theory&diff=18858Math 567 -- Elementary Number Theory2020-01-31T23:45:17Z<p>Andreic: </p>
<hr />
<div>'''MATH 567<br />
<br />
Elementary Number Theory'''<br />
<br />
MWF 1:20-2:10, Van Vleck B119<br />
<br />
'''Professor:''' [http://www.math.wisc.edu/~andreic/ Andrei Caldararu] (andreic@math.wisc.edu)<br />
''Office Hours:'' Wednesdays 2:30-3:30, Van Vleck 605. <br />
<br />
'''Grader:''' TBA<br />
''Office Hours:'' TBA.<br />
<br />
Math 567 is a course in elementary number theory, aimed at undergraduates majoring in math or other quantitative disciplines. A general familiarity with abstract algebra at the level of Math 541 will be assumed, but students who haven't taken 541 are welcome to attend if they're willing to play a little catchup. We will be using William Stein's new (and cheap) textbook [http://www.amazon.com/Elementary-Number-Theory-Computational-Undergraduate/dp/0387855246 Elementary Number Theory: Primes, Congruences, and Secrets], which emphasizes computational approaches to the subject. If you don't need a physical copy of the book, [http://www.williamstein.org/ent/ it is available as a free legal .pdf.] We will be using the (free, public-domain) mathematical software [http://www.sagemath.org/ SAGE], developed largely by Stein, as an integral component of our coursework. There is a [http://doc.sagemath.org/pdf/en/tutorial/SageTutorial.pdf useful online tutorial.] You can download SAGE to your own computer or [http://www.sagenb.org use it online].<br />
<br />
Topics include some subset of, but are not limited to: Divisibility, the Euclidean algorithm and the GCD, linear Diophantine equations, prime numbers and uniqueness of factorization. Congruences, Chinese remainder theorem, Fermat's "little" theorem, Wilson's theorem, Euler's theorem and totient function, the RSA cryptosystem, Rabin's encryption scheme, Diffie-Hellman key exchange protocol. Number-theoretic functions, multiplicative functions, Möbius inversion. Primitive roots and indices. Quadratic reciprocity and the Legendre symbol. Perfect numbers, Mersenne primes, Fermat primes. Pythagorean triples, Special cases of Fermat's "last" theorem. Fibonacci numbers. Continued fractions. Distribution of primes, discussion of prime number theorem. Primality testing and factoring algorithms.<br />
<br />
'''Course Policies:''' Homework will be due on Fridays. It can be turned in late only with ''advance'' permission from your grader. It is acceptable to use calculators and computers on homework (indeed, some of it will require a computer) but calculators are not allowed during exams. You are encouraged to work together on homework, but writeups must be done individually.<br />
<br />
Many of the problems in this course will ask you to prove things. I expect proofs to be written in English sentences; the proofs in Stein's book are a good model for the level of verbosity I am looking for. <br />
<br />
'''Grading:''' The grade in Math 567 will be composed of 50% homework, 25% midterm, 25% final. The midterm will be on March 11, in class. The final exam will be on 5/8/2020, 7:45AM-9:45AM in room TBA.<br />
<br />
'''Syllabus:''' <br />
(This may change as we see what pace works well for the course. All section numbers refer to Stein's book.)<br />
<br />
* Jan 22-31: Prime numbers, prime factorizations, Euclidean algorithm and GCD (1.1-1.2)<br />
* Feb 3-7: The integers mod n, Euler's theorem, the phi function (2.1-2.2)<br />
* Feb 10-14: Modular exponentiation, primality testing, and primitive roots (2.4-2.5)<br />
* Feb 17-21: Public-key cryptography and RSA (3.1-3.4)<br />
* Feb 24-28: Rabin's algorithm (not in the book); algebraic numbers <br />
* Mar 2-6: Quadratic reciprocity (4.1-4.4)<br />
* Mar 9, Mar 13: Finite and infinite continued fractions (5.1-5.3)<br />
* Mar 11: ''Midterm exam''<br />
* Mar 23-30: Continued fractions and diophantine approximation (5.4-5.5)<br />
* Apr 1-3: Diophantine equations I: Pell's equation and Lagrange's theorem<br />
* Apr 6-10: More diophantine equations, elliptic curves (6.1)<br />
* Apr 13-17: Applications of elliptic curves (6.2-6.3)<br />
* Apr 20-24: More applications of elliptic curves (6.4)<br />
* Apr 27-May 1: advanced topic TBD: maybe additional discussion of cryptographic techniques?<br />
<br />
'''Homework:'''<br />
Homework is due at the beginning of class on the specified Friday. Typing your homework is not a requirement, but if you don't already know LaTeX I highly recommend that you learn it and use it to typeset your homework. I will sometimes assign extra problems, which I will e-mail to the class list and include here.<br />
<br />
* '''Jan 31''': 1.1, 1.3, 1.5, 1.7 (use SAGE), 1.8, 1.14.<br />
Problem A: Use SAGE to compute the number of x in [1..N] such that x^2 + 1 is prime, for N = 100, N = 1000, and N = 10000. Let f(N) be the number of such x. <br />
<br />
a) Can you formulate a conjecture about the relationship between f(N) and N/log N? <br />
<br />
b) What if x^2 + 1 is replaced with x^2 + 2? Can you explain why x^2 + 2 appears less likely to be prime? (Hint: consider x mod 3.)<br />
<br />
c) Prove that f(N) is at most (1/2)N+1. (Hint: consider x mod 2.) <br />
<br />
d) Give as good an upper bound as you can for f(N).<br />
<br />
Note that, despite the evident regularities you'll observe in this problem, we do not even know whether there are infinitely many primes of the form x^2 + 1! You would become very famous if you proved this.<br />
<br />
* Feb 7: 2.6 (the formulation of numerical evidence should be done by Sage if you've got Sage working, and by calculator if not; you can use an online tool like [http://primes.utm.edu/curios/includes/primetest.php this] to test whether a number is prime.) 2.8,2.9,2.11,2.12,2.14,2.19<br />
<br />
<!--<br />
* '''Oct 5''': 2.15, 2.18, 2.23, 2.26, 2.30.<br />
<br />
Problem A: Prove that if n=pq, with p,q prime, then n is not a Carmichael number.<br />
<br />
* '''Oct 12''': Book problems: 3.4, 3.5, 3.6<br />
<br />
Problem A. Prove that there are infinitely many primes p such that 2 is '''not''' a primitive root in Z/pZ. We break this up into steps.<br />
<br />
Problem A.1. Prove that, if x is an element of Z/nZ, then x^2 is not a primitive root.<br />
<br />
Problem A.2. Prove that there are infinitely many primes p such that 2 is a square in Z/pZ. Hint: suppose there are only finitely many such primes p_1, .. p_r, and define N = (p_1 .. p_r)^2 - 2. Where can you go from here...?<br />
<br />
Problem A.3. Give a list of five primes p such that 2 is not a primitive root in Z/pZ (you can use the method of this proof or any other.)<br />
<br />
Problem B. Prove that 24 is the largest integer n such that every element of (Z/nZ)^* is a root of x^2-1.<br />
<br />
* '''Oct 19''': Problem A. Using p = 23 and q=31, show how to encrypt the message x=240 with Rabin's algorithm. Find all possible decryptions of your encrypted message.<br />
<br />
Problem B. Give a prime factorization of the Gaussian integer 7+9i.<br />
<br />
Problem C. Read the notes from [http://www.math.ucsd.edu/~jverstra/Gaussian1.pdf here]. Note that Z[i] satisfies a ''reduction theorem'': if n and d are Gaussian integers, then there exists integers q and r such that n = qd + r and Norm(r) < Norm(d). But (by contrast with the case of Z) this d may not be unique. In some contexts it is better to be able to choose r uniquely, even if this means letting r have norm greater than Norm(d). <br />
<br />
C.1. When d = 1+2i, show that, for each n in Z[i], there is a '''unique''' pair (q,r) in Z[i] such that n = qd+r and r is contained in the set {0,1,2,3,4}. For instance, i can be written as i(1+2i) + 2, so we say i reduces to 2 mod (1+2i).<br />
(Hint: Suppose q(1+2i)+r = q'(1+2i) + r'. What can we say about (r-r'), and why is this incompatible with both r and r' being in {0,1,2,3,4}?<br />
<br />
C.2. Show that if n is an integer in Z, the reduction of n mod (1+2i) is equal to its reduction mod 5.<br />
<br />
Problem D. Let's try to figure out how to define "phi(d)" for a Gaussian integer d. Suppose S is a set of Gaussian integers such that every n in Z[i] can be written uniquely as qd+r, with q in Z[i] and r in S. (So for instance when d=1+2i, we showed in problem B that we can take S to be {0...4}. It would also be OK to take S to be {1..5} or {0,i,2i,1+i,1+2i}. In fact, it turns out that S has to be a set of size Norm(d) (I might or might not prove this in class; if not, feel free just to accept it.)<br />
<br />
Now define phi(d) to be the number of elements s of S such that s and d are coprime.<br />
<br />
D.1. Compute phi(1+2i) and phi(3).<br />
<br />
D.2. Prove that the value of phi(d) does not depend on the choice of S.<br />
<br />
D.3. Prove that every n in Z[i] which is coprime to 3 satisfies n^phi(3) = 1 mod 3; that is, Euler's theorem holds in this case. (You can prove this by direct computation; of course, if you want, you are welcome to prove that Euler's theorem holds for Z[i] in general, adapting the proof in Stein's book or the one we gave in class.)<br />
<br />
*'''Nov 2:''' 4.1 from the book.<br />
<br />
Problem A. Express 50005 as the sum of two squares.<br />
<br />
In the next two problems we denote by r(n) the number of ways to express n as the sum of two squares (i.e. the number of pairs (a,b) such that a^2 + b^2 = n.) For instance, r(5) = 8 (as shown on the midterm.)<br />
<br />
Problem B. Prove that, for any N, there exists an integer n such that r(n) > N. (I.E., the function r(n) is ''unbounded.''<br />
<br />
Problem C. If you like Sage, write a short program in Sage to compute r(n) and compute the average of r(n) as n ranges from 1 to 1000. Whether or not you like Sage, make a guess as to how this average would behave if you replaced "1000" by a larger and larger number. (Feel free to ask the Sage-lovers what answer they got in the optional first part of the question.) Can you prove this guess is correct?<br />
<br />
Problem D. We saw in class that the ring Z[sqrt(-5)] doesn't have unique factorization; 6 can be factored as 2*3 or (1+sqrt(-5))(1-sqrt(-5)). In this problem, we will prove that Z[sqrt(-d)] fails to have unique factorization for EVERY odd d >= 5. (Actually it's true for all d >= 5 but to make the proof manageable we'll restrict to the odd case.]<br />
<br />
D.1. Show that (1+sqrt(-d)), (1-sqrt(-d)) and 2 are irreducible in Z[sqrt(-d)].<br />
<br />
D.2. Now give an element of Z[sqrt(-d)] that has two distinct factorizations into irreducibles. (Hint: imitate the example we used for Z[sqrt(-5)].)<br />
''Remark:'' The rings Z[sqrt(d)], where d is positive, are quite different -- here we believe that there are infinitely many with unique factorization, though this conjecture has remained unproved for many decades!<br />
<br />
*'''November 16'''<br />
<br />
Book problems: 5.3,5.4,5.5<br />
<br />
Problem A: We discussed in class the problem of studying which positive integers are the sum of two squares. In this problem we prove that every element n of (Z/pZ) is the sum of two squares. We argue as follows. Let S be the set of squares in (Z/pZ), and let T be the set of (Z/pZ) consisting of all elements of the form n - x^2, for some x in (Z/pZ).<br />
<br />
A.1. What is the size of S and of T? Use this to show that S and T are not disjoint.<br />
<br />
A.2. Given that S and T are not disjoint, prove that n is the sum of two squares.<br />
<br />
Problem B. Using the continued fraction expansion, find a solution to the Pell equation x^2 - 13 y^2 = 1.<br />
<br />
Problem C. Show that the modified Pell equation x^2 - 7y^2 = -1 has no solutions in integers x,y. (Hint: reduce the equation modulo a suitably chosen prime.)<br />
<br />
Problem D. Stuffy Stirnweiss finished the 1945 season with a batting average of .3085443. Using continued fractions, guess how many at-bats he had. Tony Cuccinello had a batting average of .3084577. Given that he had more than 200 and fewer than 600 at-bats, can you estimate the number of at-bats he had?<br />
<br />
'''December 7'''<br />
<br />
Problem A. When p is a prime congruent to 3 mod 4, prove that ((p-1)/2)! is either 1 or -1 in (Z/pZ). '''OPTIONAL:''' Use sage to compute whether this factorial is 1 or -1 for many primes p. Is there any pattern? Does it seem to be 1 half the time and -1 half the time? Note that I have no idea what the answer to this question is.<br />
<br />
Problem B. Using the continued fraction expansion, find a solution to the Pell equation x^2 - 13 y^2 = 1.<br />
<br />
Problem C. Show that the modified Pell equation x^2 - 7y^2 = -1 has no solutions in integers x,y. (Hint: reduce the equation modulo a suitably chosen prime.)<br />
<br />
Problem D.1. Pick two values of a in F_11 = Z/11Z (a not equal to 3), such that the equation y^2 = x^3+ax+1 defines an elliptic curve (i.e., it is smooth). For each such a, determine the number of points #E(F_11) and check that it falls inside the interval described in class. <br />
<br />
Problem D.2. The point P = (0,1) lies on each of these curves. For a =3 (the curve discussed in class) determine the order of P in the elliptic curve group, that is, find the smallest positive integer n such that nP = (infinity) -- the identity element in the group law.<br />
<br />
<!-- '''November 19'''<br />
<br />
Problem A. Using the method discussed in class (which is also the method of problem 4.6 in Stein, which in retrospect I think was too hard to assign with no preparation) find a nontrivial solution with y > 0 to the equation x^2 + 11y^2 = z^2. <br />
Problem B. Let f(X) be the number of solutions of A^2 + B^2 = C^3 such that A^2, B^2, and C^3 are all at most X. Use the heuristic described in class to explain why one might expect f(X) to grow more or less like X^{1/3}.<br />
<br />
Problem C. Show that if A+Bi is the cube of a Gaussian integer, then A^2 + B^2 is a perfect cube.<br />
<br />
Do ONE of problem D1 and D2; D1 is for people who like Sage, D2 is for people who like proving things.<br />
<br />
D1. Use Sage to compute f(X) for X = 1000, 10000, 100000. Does the answer look consistent with the heuristic prediction you made in B? Does f(X)/X^{1/3} appear to be approaching a limit?<br />
D2. We will use problem C to give a lower bound for f(X). Use this to show that there are at least X^{1/3} Gaussian integers with norm at most X that are perfect cubes. From here, show that f(X) > X^{1/3}.<br />
<br />
OPTIONAL: Show that the converse of problem C also holds -- A+Bi has norm a perfect cube if and only if A+Bi is a perfect cube. (You will need unique factorization of Gaussian integers.) Using this fact, prove that f(X) is bounded above by C X^{1/3} for some constant C.<br />
<br />
'''December 8 (note nonstandard due date)'''<br />
<br />
Problem A: Hyperbolas, ellipses, and "magic slopes"<br />
<br />
In the usual analytic geometry, both hyperbolas and ellipses are given by equations of the form<br />
<br />
ax^2 + bxy + cy^2 = d (*)<br />
<br />
<br />
(we always assume d is nonzero.)<br />
<br />
How can we tell whether such an equation describes an ellipse or a<br />
hyperbola? Well, in the geometric setting we all know and love (i.e. in<br />
R^2) a hyperbola has asymptotes and an ellipse does not. What is an <br />
asymptote? You could say it's "a line which doesn't intersect the curve", <br />
but ellipses have such lines too. You might want to say "a line which <br />
doesn't intersect the curve but comes closer and closer to the curve," but <br />
the problem here is that this a) is somewhat imprecise, and b) relies on a <br />
notion of "closer" that is not going to be very clear in Z/pZ, which is <br />
our ultimate goal!<br />
<br />
<br />
So let me state it a different way: let's say the asymptotes of a <br />
hyperbola have slopes m_1 and m_2. These are "magic slopes" in the <br />
following sense: ANY line of slope m_1 (and ditto for m_2) intersects the <br />
hyperbola in at most one point. An ellipse doesn't have "magic slopes" -- <br />
you can convince yourself by drawing pictures that, for any slope m, you <br />
can find a line of slope m striking the ellipse twice.<br />
<br />
<br />
[REMARK: For technical reasons we're going to ignore hyperbolae with a <br />
vertical asymptote, since vertical lines don't exactly have slope.]<br />
<br />
<br />
Now you might know the criterion that, over the real numbers.<br />
<br />
<br />
ax^2 + bxy + cy^2 = d<br />
<br />
<br />
is a hyperbola if b^2 - 4ac > 0 and an ellipse if b^2 - 4ac < 0. (For <br />
instance, x^2 - y^2 = 1 is a hyperbola with magic slopes 1 and -1, and x^2 <br />
+ y^2 = 1 is an ellipse.) <br />
This criterion does not make sense in Z/pZ, where there is no notion of <br />
"greater" or "less."<br />
<br />
<br />
QUESTION A.1.<br />
<br />
For which primes p is it the case that <br />
<br />
x^2 + y^2 = 1<br />
<br />
has magic slopes in Z/pZ? <br />
<br />
QUESTION A.2. For which primes p is it the case that<br />
<br />
<br />
x^2 - y^2 = 1<br />
<br />
<br />
has magic slopes in Z/pZ?<br />
<br />
<br />
QUESTION B: Back in the very first week of this course we studied the set of integers x such that x^2 + 1 is prime. Based on the heuristics we discussed in class, about how many x between 1 and N would you expect to satisfy "x^2 + 1 is prime?"<br />
<br />
<br />
QUESTION C: Let z be a complex number. Then the set of all complex numbers of the form a + bz, with a and b integers, forms a lattice in the complex plane (thought of as R^2.) What is the covolume of this lattice, in terms of z?<br />
<br />
<br />
QUESTION D: Let z be a complex number. Using Minkowski's theorem and the result of question C, show that there exist integers a,b, not both zero, such that Norm(a+bz) is at most (4/pi) Im(z). (Hint: let Omega be the set of all complex numbers of norm at most (4/pi) Im(z).)<br />
<br />
<br />
OPTIONAL EXTRA: Can you give an exact formula for the minimal norm of any nonzero complex number of the form a+bz?<br />
<br />
'''December 12''' Book problems 6.1, 6.2, 6.5, 6.10. For extra credit attempt to do problem 6.9.<br />
<br />
--></div>Andreichttps://www.math.wisc.edu/wiki/index.php?title=Algebra_and_Algebraic_Geometry_Seminar_Spring_2020&diff=18710Algebra and Algebraic Geometry Seminar Spring 20202020-01-20T18:15:04Z<p>Andreic: </p>
<hr />
<div>== Spring 2020 Schedule ==<br />
<br />
{| cellpadding="8"<br />
!align="left" | date<br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s) <br />
|-<br />
|January 24<br />
|[http://www.math.ualberta.ca/~xichen// Xi Chen (Alberta)]<br />
|[[#Xi Chen|Rational Curves on K3 Surfaces]]<br />
|Michael K<br />
|-<br />
|January 31<br />
|[http://www.math.utah.edu/~letz// Janina Letz (Utah)]<br />
|TBD<br />
|Daniel and Michael B<br />
|-<br />
|February 7<br />
|Jonathan Monta&#241;o (New Mexico State)<br />
|TBD<br />
|Daniel<br />
|-<br />
|February 14<br />
|<br />
|<br />
| <br />
|-<br />
|February 21<br />
|Erika Ordog (Duke)<br />
|TBD<br />
|Daniel<br />
|-<br />
|February 28<br />
|<br />
|<br />
|<br />
|-<br />
|March 6<br />
|<br />
|<br />
|<br />
|-<br />
|March 13<br />
|<br />
|<br />
|<br />
|-<br />
|March 20<br />
|<br />
|<br />
|<br />
|-<br />
|March 27<br />
|[https://mcfaddin.github.io// Patrick McFaddin (Fordham)]<br />
|TBD<br />
|Michael B<br />
|-<br />
|April 3<br />
|<br />
| <br />
| <br />
|-<br />
|April 10<br />
|[https://sites.google.com/view/ruijie-yang/ Ruijie Yang (Stony Brook)]<br />
|TBD<br />
|Michael K<br />
|-<br />
|April 17<br />
|Remy van Dobben de Bruyn (Princeton/IAS)<br />
|TBD<br />
|Botong<br />
|-<br />
|April 24<br />
|Katrina Honigs (University of Oregon)<br />
|TBA<br />
|Andrei<br />
|-<br />
|May 1<br />
|Lazarsfeld Distinguished Lectures<br />
|<br />
|<br />
|-<br />
|May 8<br />
|<br />
|<br />
| <br />
|-<br />
|}<br />
<br />
== Abstracts ==<br />
===Xi Chen===<br />
'''Rational Curves on K3 Surfaces<br />
'''<br />
<br />
It is conjectured that there are infinitely many rational<br />
curves on every projective K3 surface. A large part of this conjecture<br />
was proved by Jun Li and Christian Liedtke, based on the<br />
characteristic p reduction method proposed by<br />
Bogomolov-Hassett-Tschinkel. They proved that there are infinitely<br />
many rational curves on every projective K3 surface of odd Picard<br />
rank. Over complex numbers, there are a few remaining cases: K3<br />
surfaces of Picard rank two excluding elliptic K3's and K3's with<br />
infinite automorphism groups and K3 surfaces with two particular<br />
Picard lattices of rank four. We have settled these leftover cases and also<br />
generalized the conjecture to the existence of curves of high genus.<br />
This is a joint work with Frank Gounelas and Christian Liedtke.<br />
<br />
===Janina Letz===<br />
'''Local to global principles for generation time over commutative<br />
rings<br />
'''<br />
<br />
Abstract: In the derived category of modules over a commutative<br />
noetherian ring a complex $G$ is said to generate a complex $X$ if the<br />
latter can be obtained from the former by taking finitely many summands<br />
and cones. The number of cones needed in this process is the generation<br />
time of $X$. In this talk I will present some local to global type<br />
results for computing this invariant, and also discuss some<br />
applications of these results.</div>Andreichttps://www.math.wisc.edu/wiki/index.php?title=Algebra_and_Algebraic_Geometry_Seminar&diff=18709Algebra and Algebraic Geometry Seminar2020-01-20T18:13:41Z<p>Andreic: Redirected page to Algebra and Algebraic Geometry Seminar Spring 2020</p>
<hr />
<div>#REDIRECT [[Algebra and Algebraic Geometry Seminar Spring 2020]]</div>Andreichttps://www.math.wisc.edu/wiki/index.php?title=Math_567_--_Elementary_Number_Theory&diff=18696Math 567 -- Elementary Number Theory2020-01-19T17:19:39Z<p>Andreic: </p>
<hr />
<div>'''MATH 567<br />
<br />
Elementary Number Theory'''<br />
<br />
MWF 1:20-2:10, Van Vleck B119<br />
<br />
'''Professor:''' [http://www.math.wisc.edu/~andreic/ Andrei Caldararu] (andreic@math.wisc.edu)<br />
''Office Hours:'' Wednesdays 2:30-3:30, Van Vleck 605. <br />
<br />
'''Grader:''' TBA<br />
''Office Hours:'' TBA.<br />
<br />
Math 567 is a course in elementary number theory, aimed at undergraduates majoring in math or other quantitative disciplines. A general familiarity with abstract algebra at the level of Math 541 will be assumed, but students who haven't taken 541 are welcome to attend if they're willing to play a little catchup. We will be using William Stein's new (and cheap) textbook [http://www.amazon.com/Elementary-Number-Theory-Computational-Undergraduate/dp/0387855246 Elementary Number Theory: Primes, Congruences, and Secrets], which emphasizes computational approaches to the subject. If you don't need a physical copy of the book, [http://www.williamstein.org/ent/ it is available as a free legal .pdf.] We will be using the (free, public-domain) mathematical software [http://www.sagemath.org/ SAGE], developed largely by Stein, as an integral component of our coursework. There is a [http://doc.sagemath.org/pdf/en/tutorial/SageTutorial.pdf useful online tutorial.] You can download SAGE to your own computer or [http://www.sagenb.org use it online].<br />
<br />
Topics include some subset of, but are not limited to: Divisibility, the Euclidean algorithm and the GCD, linear Diophantine equations, prime numbers and uniqueness of factorization. Congruences, Chinese remainder theorem, Fermat's "little" theorem, Wilson's theorem, Euler's theorem and totient function, the RSA cryptosystem, Rabin's encryption scheme, Diffie-Hellman key exchange protocol. Number-theoretic functions, multiplicative functions, Möbius inversion. Primitive roots and indices. Quadratic reciprocity and the Legendre symbol. Perfect numbers, Mersenne primes, Fermat primes. Pythagorean triples, Special cases of Fermat's "last" theorem. Fibonacci numbers. Continued fractions. Distribution of primes, discussion of prime number theorem. Primality testing and factoring algorithms.<br />
<br />
'''Course Policies:''' Homework will be due on Fridays. It can be turned in late only with ''advance'' permission from your grader. It is acceptable to use calculators and computers on homework (indeed, some of it will require a computer) but calculators are not allowed during exams. You are encouraged to work together on homework, but writeups must be done individually.<br />
<br />
Many of the problems in this course will ask you to prove things. I expect proofs to be written in English sentences; the proofs in Stein's book are a good model for the level of verbosity I am looking for. <br />
<br />
'''Grading:''' The grade in Math 567 will be composed of 50% homework, 25% midterm, 25% final. The midterm will be on March 11, in class. The final exam will be on 5/8/2020, 7:45AM-9:45AM in room TBA.<br />
<br />
'''Syllabus:''' <br />
(This may change as we see what pace works well for the course. All section numbers refer to Stein's book.)<br />
<br />
* Jan 22-31: Prime numbers, prime factorizations, Euclidean algorithm and GCD (1.1-1.2)<br />
* Feb 3-7: The integers mod n, Euler's theorem, the phi function (2.1-2.2)<br />
* Feb 10-14: Modular exponentiation, primality testing, and primitive roots (2.4-2.5)<br />
* Feb 17-21: Public-key cryptography and RSA (3.1-3.4)<br />
* Feb 24-28: Rabin's algorithm (not in the book); algebraic numbers <br />
* Mar 2-6: Quadratic reciprocity (4.1-4.4)<br />
* Mar 9, Mar 13: Finite and infinite continued fractions (5.1-5.3)<br />
* Mar 11: ''Midterm exam''<br />
* Mar 23-30: Continued fractions and diophantine approximation (5.4-5.5)<br />
* Apr 1-3: Diophantine equations I: Pell's equation and Lagrange's theorem<br />
* Apr 6-10: More diophantine equations, elliptic curves (6.1)<br />
* Apr 13-17: Applications of elliptic curves (6.2-6.3)<br />
* Apr 20-24: More applications of elliptic curves (6.4)<br />
* Apr 27-May 1: advanced topic TBD: maybe additional discussion of cryptographic techniques?<br />
<br />
'''Homework:'''<br />
Homework is due at the beginning of class on the specified Friday. Typing your homework is not a requirement, but if you don't already know LaTeX I highly recommend that you learn it and use it to typeset your homework. I will sometimes assign extra problems, which I will e-mail to the class list and include here.<br />
<br />
* '''Jan 31''': 1.1, 1.3, 1.5, 1.7 (use SAGE), 1.8, 1.14.<br />
Problem A: Use SAGE to compute the number of x in [1..N] such that x^2 + 1 is prime, for N = 100, N = 1000, and N = 10000. Let f(N) be the number of such x. <br />
<br />
a) Can you formulate a conjecture about the relationship between f(N) and N/log N? <br />
<br />
b) What if x^2 + 1 is replaced with x^2 + 2? Can you explain why x^2 + 2 appears less likely to be prime? (Hint: consider x mod 3.)<br />
<br />
c) Prove that f(N) is at most (1/2)N+1. (Hint: consider x mod 2.) <br />
<br />
d) Give as good an upper bound as you can for f(N).<br />
<br />
Note that, despite the evident regularities you'll observe in this problem, we do not even know whether there are infinitely many primes of the form x^2 + 1! You would become very famous if you proved this.<br />
<br />
<!--<br />
* '''Sep 28''': 2.6 (the formulation of numerical evidence should be done by Sage if you've got Sage working, and by calculator if not; you can use an online tool like [http://primes.utm.edu/curios/includes/primetest.php this] to test whether a number is prime.) 2.8,2.9,2.11,2.12,2.14,2.19<br />
<br />
* '''Oct 5''': 2.15, 2.18, 2.23, 2.26, 2.30.<br />
<br />
Problem A: Prove that if n=pq, with p,q prime, then n is not a Carmichael number.<br />
<br />
* '''Oct 12''': Book problems: 3.4, 3.5, 3.6<br />
<br />
Problem A. Prove that there are infinitely many primes p such that 2 is '''not''' a primitive root in Z/pZ. We break this up into steps.<br />
<br />
Problem A.1. Prove that, if x is an element of Z/nZ, then x^2 is not a primitive root.<br />
<br />
Problem A.2. Prove that there are infinitely many primes p such that 2 is a square in Z/pZ. Hint: suppose there are only finitely many such primes p_1, .. p_r, and define N = (p_1 .. p_r)^2 - 2. Where can you go from here...?<br />
<br />
Problem A.3. Give a list of five primes p such that 2 is not a primitive root in Z/pZ (you can use the method of this proof or any other.)<br />
<br />
Problem B. Prove that 24 is the largest integer n such that every element of (Z/nZ)^* is a root of x^2-1.<br />
<br />
* '''Oct 19''': Problem A. Using p = 23 and q=31, show how to encrypt the message x=240 with Rabin's algorithm. Find all possible decryptions of your encrypted message.<br />
<br />
Problem B. Give a prime factorization of the Gaussian integer 7+9i.<br />
<br />
Problem C. Read the notes from [http://www.math.ucsd.edu/~jverstra/Gaussian1.pdf here]. Note that Z[i] satisfies a ''reduction theorem'': if n and d are Gaussian integers, then there exists integers q and r such that n = qd + r and Norm(r) < Norm(d). But (by contrast with the case of Z) this d may not be unique. In some contexts it is better to be able to choose r uniquely, even if this means letting r have norm greater than Norm(d). <br />
<br />
C.1. When d = 1+2i, show that, for each n in Z[i], there is a '''unique''' pair (q,r) in Z[i] such that n = qd+r and r is contained in the set {0,1,2,3,4}. For instance, i can be written as i(1+2i) + 2, so we say i reduces to 2 mod (1+2i).<br />
(Hint: Suppose q(1+2i)+r = q'(1+2i) + r'. What can we say about (r-r'), and why is this incompatible with both r and r' being in {0,1,2,3,4}?<br />
<br />
C.2. Show that if n is an integer in Z, the reduction of n mod (1+2i) is equal to its reduction mod 5.<br />
<br />
Problem D. Let's try to figure out how to define "phi(d)" for a Gaussian integer d. Suppose S is a set of Gaussian integers such that every n in Z[i] can be written uniquely as qd+r, with q in Z[i] and r in S. (So for instance when d=1+2i, we showed in problem B that we can take S to be {0...4}. It would also be OK to take S to be {1..5} or {0,i,2i,1+i,1+2i}. In fact, it turns out that S has to be a set of size Norm(d) (I might or might not prove this in class; if not, feel free just to accept it.)<br />
<br />
Now define phi(d) to be the number of elements s of S such that s and d are coprime.<br />
<br />
D.1. Compute phi(1+2i) and phi(3).<br />
<br />
D.2. Prove that the value of phi(d) does not depend on the choice of S.<br />
<br />
D.3. Prove that every n in Z[i] which is coprime to 3 satisfies n^phi(3) = 1 mod 3; that is, Euler's theorem holds in this case. (You can prove this by direct computation; of course, if you want, you are welcome to prove that Euler's theorem holds for Z[i] in general, adapting the proof in Stein's book or the one we gave in class.)<br />
<br />
*'''Nov 2:''' 4.1 from the book.<br />
<br />
Problem A. Express 50005 as the sum of two squares.<br />
<br />
In the next two problems we denote by r(n) the number of ways to express n as the sum of two squares (i.e. the number of pairs (a,b) such that a^2 + b^2 = n.) For instance, r(5) = 8 (as shown on the midterm.)<br />
<br />
Problem B. Prove that, for any N, there exists an integer n such that r(n) > N. (I.E., the function r(n) is ''unbounded.''<br />
<br />
Problem C. If you like Sage, write a short program in Sage to compute r(n) and compute the average of r(n) as n ranges from 1 to 1000. Whether or not you like Sage, make a guess as to how this average would behave if you replaced "1000" by a larger and larger number. (Feel free to ask the Sage-lovers what answer they got in the optional first part of the question.) Can you prove this guess is correct?<br />
<br />
Problem D. We saw in class that the ring Z[sqrt(-5)] doesn't have unique factorization; 6 can be factored as 2*3 or (1+sqrt(-5))(1-sqrt(-5)). In this problem, we will prove that Z[sqrt(-d)] fails to have unique factorization for EVERY odd d >= 5. (Actually it's true for all d >= 5 but to make the proof manageable we'll restrict to the odd case.]<br />
<br />
D.1. Show that (1+sqrt(-d)), (1-sqrt(-d)) and 2 are irreducible in Z[sqrt(-d)].<br />
<br />
D.2. Now give an element of Z[sqrt(-d)] that has two distinct factorizations into irreducibles. (Hint: imitate the example we used for Z[sqrt(-5)].)<br />
''Remark:'' The rings Z[sqrt(d)], where d is positive, are quite different -- here we believe that there are infinitely many with unique factorization, though this conjecture has remained unproved for many decades!<br />
<br />
*'''November 16'''<br />
<br />
Book problems: 5.3,5.4,5.5<br />
<br />
Problem A: We discussed in class the problem of studying which positive integers are the sum of two squares. In this problem we prove that every element n of (Z/pZ) is the sum of two squares. We argue as follows. Let S be the set of squares in (Z/pZ), and let T be the set of (Z/pZ) consisting of all elements of the form n - x^2, for some x in (Z/pZ).<br />
<br />
A.1. What is the size of S and of T? Use this to show that S and T are not disjoint.<br />
<br />
A.2. Given that S and T are not disjoint, prove that n is the sum of two squares.<br />
<br />
Problem B. Using the continued fraction expansion, find a solution to the Pell equation x^2 - 13 y^2 = 1.<br />
<br />
Problem C. Show that the modified Pell equation x^2 - 7y^2 = -1 has no solutions in integers x,y. (Hint: reduce the equation modulo a suitably chosen prime.)<br />
<br />
Problem D. Stuffy Stirnweiss finished the 1945 season with a batting average of .3085443. Using continued fractions, guess how many at-bats he had. Tony Cuccinello had a batting average of .3084577. Given that he had more than 200 and fewer than 600 at-bats, can you estimate the number of at-bats he had?<br />
<br />
'''December 7'''<br />
<br />
Problem A. When p is a prime congruent to 3 mod 4, prove that ((p-1)/2)! is either 1 or -1 in (Z/pZ). '''OPTIONAL:''' Use sage to compute whether this factorial is 1 or -1 for many primes p. Is there any pattern? Does it seem to be 1 half the time and -1 half the time? Note that I have no idea what the answer to this question is.<br />
<br />
Problem B. Using the continued fraction expansion, find a solution to the Pell equation x^2 - 13 y^2 = 1.<br />
<br />
Problem C. Show that the modified Pell equation x^2 - 7y^2 = -1 has no solutions in integers x,y. (Hint: reduce the equation modulo a suitably chosen prime.)<br />
<br />
Problem D.1. Pick two values of a in F_11 = Z/11Z (a not equal to 3), such that the equation y^2 = x^3+ax+1 defines an elliptic curve (i.e., it is smooth). For each such a, determine the number of points #E(F_11) and check that it falls inside the interval described in class. <br />
<br />
Problem D.2. The point P = (0,1) lies on each of these curves. For a =3 (the curve discussed in class) determine the order of P in the elliptic curve group, that is, find the smallest positive integer n such that nP = (infinity) -- the identity element in the group law.<br />
<br />
<!-- '''November 19'''<br />
<br />
Problem A. Using the method discussed in class (which is also the method of problem 4.6 in Stein, which in retrospect I think was too hard to assign with no preparation) find a nontrivial solution with y > 0 to the equation x^2 + 11y^2 = z^2. <br />
Problem B. Let f(X) be the number of solutions of A^2 + B^2 = C^3 such that A^2, B^2, and C^3 are all at most X. Use the heuristic described in class to explain why one might expect f(X) to grow more or less like X^{1/3}.<br />
<br />
Problem C. Show that if A+Bi is the cube of a Gaussian integer, then A^2 + B^2 is a perfect cube.<br />
<br />
Do ONE of problem D1 and D2; D1 is for people who like Sage, D2 is for people who like proving things.<br />
<br />
D1. Use Sage to compute f(X) for X = 1000, 10000, 100000. Does the answer look consistent with the heuristic prediction you made in B? Does f(X)/X^{1/3} appear to be approaching a limit?<br />
D2. We will use problem C to give a lower bound for f(X). Use this to show that there are at least X^{1/3} Gaussian integers with norm at most X that are perfect cubes. From here, show that f(X) > X^{1/3}.<br />
<br />
OPTIONAL: Show that the converse of problem C also holds -- A+Bi has norm a perfect cube if and only if A+Bi is a perfect cube. (You will need unique factorization of Gaussian integers.) Using this fact, prove that f(X) is bounded above by C X^{1/3} for some constant C.<br />
<br />
'''December 8 (note nonstandard due date)'''<br />
<br />
Problem A: Hyperbolas, ellipses, and "magic slopes"<br />
<br />
In the usual analytic geometry, both hyperbolas and ellipses are given by equations of the form<br />
<br />
ax^2 + bxy + cy^2 = d (*)<br />
<br />
<br />
(we always assume d is nonzero.)<br />
<br />
How can we tell whether such an equation describes an ellipse or a<br />
hyperbola? Well, in the geometric setting we all know and love (i.e. in<br />
R^2) a hyperbola has asymptotes and an ellipse does not. What is an <br />
asymptote? You could say it's "a line which doesn't intersect the curve", <br />
but ellipses have such lines too. You might want to say "a line which <br />
doesn't intersect the curve but comes closer and closer to the curve," but <br />
the problem here is that this a) is somewhat imprecise, and b) relies on a <br />
notion of "closer" that is not going to be very clear in Z/pZ, which is <br />
our ultimate goal!<br />
<br />
<br />
So let me state it a different way: let's say the asymptotes of a <br />
hyperbola have slopes m_1 and m_2. These are "magic slopes" in the <br />
following sense: ANY line of slope m_1 (and ditto for m_2) intersects the <br />
hyperbola in at most one point. An ellipse doesn't have "magic slopes" -- <br />
you can convince yourself by drawing pictures that, for any slope m, you <br />
can find a line of slope m striking the ellipse twice.<br />
<br />
<br />
[REMARK: For technical reasons we're going to ignore hyperbolae with a <br />
vertical asymptote, since vertical lines don't exactly have slope.]<br />
<br />
<br />
Now you might know the criterion that, over the real numbers.<br />
<br />
<br />
ax^2 + bxy + cy^2 = d<br />
<br />
<br />
is a hyperbola if b^2 - 4ac > 0 and an ellipse if b^2 - 4ac < 0. (For <br />
instance, x^2 - y^2 = 1 is a hyperbola with magic slopes 1 and -1, and x^2 <br />
+ y^2 = 1 is an ellipse.) <br />
This criterion does not make sense in Z/pZ, where there is no notion of <br />
"greater" or "less."<br />
<br />
<br />
QUESTION A.1.<br />
<br />
For which primes p is it the case that <br />
<br />
x^2 + y^2 = 1<br />
<br />
has magic slopes in Z/pZ? <br />
<br />
QUESTION A.2. For which primes p is it the case that<br />
<br />
<br />
x^2 - y^2 = 1<br />
<br />
<br />
has magic slopes in Z/pZ?<br />
<br />
<br />
QUESTION B: Back in the very first week of this course we studied the set of integers x such that x^2 + 1 is prime. Based on the heuristics we discussed in class, about how many x between 1 and N would you expect to satisfy "x^2 + 1 is prime?"<br />
<br />
<br />
QUESTION C: Let z be a complex number. Then the set of all complex numbers of the form a + bz, with a and b integers, forms a lattice in the complex plane (thought of as R^2.) What is the covolume of this lattice, in terms of z?<br />
<br />
<br />
QUESTION D: Let z be a complex number. Using Minkowski's theorem and the result of question C, show that there exist integers a,b, not both zero, such that Norm(a+bz) is at most (4/pi) Im(z). (Hint: let Omega be the set of all complex numbers of norm at most (4/pi) Im(z).)<br />
<br />
<br />
OPTIONAL EXTRA: Can you give an exact formula for the minimal norm of any nonzero complex number of the form a+bz?<br />
<br />
'''December 12''' Book problems 6.1, 6.2, 6.5, 6.10. For extra credit attempt to do problem 6.9.<br />
<br />
--></div>Andreichttps://www.math.wisc.edu/wiki/index.php?title=Math_567_--_Elementary_Number_Theory&diff=18695Math 567 -- Elementary Number Theory2020-01-19T17:09:18Z<p>Andreic: </p>
<hr />
<div>'''MATH 567<br />
<br />
Elementary Number Theory'''<br />
<br />
MWF 1:20-2:10, Van Vleck B119<br />
<br />
'''Professor:''' [http://www.math.wisc.edu/~andreic/ Andrei Caldararu] (andreic@math.wisc.edu)<br />
''Office Hours:'' Wednesdays 3:30-4:30, Van Vleck 605. <br />
<br />
'''Grader:''' TBA<br />
''Office Hours:'' TBA.<br />
<br />
Math 567 is a course in elementary number theory, aimed at undergraduates majoring in math or other quantitative disciplines. A general familiarity with abstract algebra at the level of Math 541 will be assumed, but students who haven't taken 541 are welcome to attend if they're willing to play a little catchup. We will be using William Stein's new (and cheap) textbook [http://www.amazon.com/Elementary-Number-Theory-Computational-Undergraduate/dp/0387855246 Elementary Number Theory: Primes, Congruences, and Secrets], which emphasizes computational approaches to the subject. If you don't need a physical copy of the book, [http://www.williamstein.org/ent/ it is available as a free legal .pdf.] We will be using the (free, public-domain) mathematical software [http://www.sagemath.org/ SAGE], developed largely by Stein, as an integral component of our coursework. There is a [http://doc.sagemath.org/pdf/en/tutorial/SageTutorial.pdf useful online tutorial.] You can download SAGE to your own computer or [http://www.sagenb.org use it online].<br />
<br />
Topics include some subset of, but are not limited to: Divisibility, the Euclidean algorithm and the GCD, linear Diophantine equations, prime numbers and uniqueness of factorization. Congruences, Chinese remainder theorem, Fermat's "little" theorem, Wilson's theorem, Euler's theorem and totient function, the RSA cryptosystem, Rabin's encryption scheme, Diffie-Hellman key exchange protocol. Number-theoretic functions, multiplicative functions, Möbius inversion. Primitive roots and indices. Quadratic reciprocity and the Legendre symbol. Perfect numbers, Mersenne primes, Fermat primes. Pythagorean triples, Special cases of Fermat's "last" theorem. Fibonacci numbers. Continued fractions. Distribution of primes, discussion of prime number theorem. Primality testing and factoring algorithms.<br />
<br />
'''Course Policies:''' Homework will be due on Fridays. It can be turned in late only with ''advance'' permission from your grader. It is acceptable to use calculators and computers on homework (indeed, some of it will require a computer) but calculators are not allowed during exams. You are encouraged to work together on homework, but writeups must be done individually.<br />
<br />
Many of the problems in this course will ask you to prove things. I expect proofs to be written in English sentences; the proofs in Stein's book are a good model for the level of verbosity I am looking for. <br />
<br />
'''Grading:''' The grade in Math 567 will be composed of 50% homework, 25% midterm, 25% final. The midterm will be on March 11, in class. The final exam will be on 5/8/2020, 7:45AM-9:45AM in room TBA.<br />
<br />
'''Syllabus:''' <br />
(This may change as we see what pace works well for the course. All section numbers refer to Stein's book.)<br />
<br />
* Jan 22-31: Prime numbers, prime factorizations, Euclidean algorithm and GCD (1.1-1.2)<br />
* Feb 3-7: The integers mod n, Euler's theorem, the phi function (2.1-2.2)<br />
* Feb 10-14: Modular exponentiation, primality testing, and primitive roots (2.4-2.5)<br />
* Feb 17-21: Public-key cryptography and RSA (3.1-3.4)<br />
* Feb 24-28: Rabin's algorithm (not in the book); algebraic numbers <br />
* Mar 2-6: Quadratic reciprocity (4.1-4.4)<br />
* Mar 9, Mar 13: Finite and infinite continued fractions (5.1-5.3)<br />
* Mar 11: ''Midterm exam''<br />
* Mar 23-30: Continued fractions and diophantine approximation (5.4-5.5)<br />
* Apr 1-3: Diophantine equations I: Pell's equation and Lagrange's theorem<br />
* Apr 6-10: More diophantine equations, elliptic curves (6.1)<br />
* Apr 13-17: Applications of elliptic curves (6.2-6.3)<br />
* Apr 20-24: More applications of elliptic curves (6.4)<br />
* Apr 27-May 1: advanced topic TBD: maybe additional discussion of cryptographic techniques?<br />
<br />
'''Homework:'''<br />
Homework is due at the beginning of class on the specified Friday. Typing your homework is not a requirement, but if you don't already know LaTeX I highly recommend that you learn it and use it to typeset your homework. I will sometimes assign extra problems, which I will e-mail to the class list and include here.<br />
<br />
* '''Jan 31''': 1.1, 1.3, 1.5, 1.7 (use SAGE), 1.8, 1.14.<br />
Problem A: Use SAGE to compute the number of x in [1..N] such that x^2 + 1 is prime, for N = 100, N = 1000, and N = 10000. Let f(N) be the number of such x. <br />
<br />
a) Can you formulate a conjecture about the relationship between f(N) and N/log N? <br />
<br />
b) What if x^2 + 1 is replaced with x^2 + 2? Can you explain why x^2 + 2 appears less likely to be prime? (Hint: consider x mod 3.)<br />
<br />
c) Prove that f(N) is at most (1/2)N+1. (Hint: consider x mod 2.) <br />
<br />
d) Give as good an upper bound as you can for f(N).<br />
<br />
Note that, despite the evident regularities you'll observe in this problem, we do not even know whether there are infinitely many primes of the form x^2 + 1! You would become very famous if you proved this.<br />
<br />
<!--<br />
* '''Sep 28''': 2.6 (the formulation of numerical evidence should be done by Sage if you've got Sage working, and by calculator if not; you can use an online tool like [http://primes.utm.edu/curios/includes/primetest.php this] to test whether a number is prime.) 2.8,2.9,2.11,2.12,2.14,2.19<br />
<br />
* '''Oct 5''': 2.15, 2.18, 2.23, 2.26, 2.30.<br />
<br />
Problem A: Prove that if n=pq, with p,q prime, then n is not a Carmichael number.<br />
<br />
* '''Oct 12''': Book problems: 3.4, 3.5, 3.6<br />
<br />
Problem A. Prove that there are infinitely many primes p such that 2 is '''not''' a primitive root in Z/pZ. We break this up into steps.<br />
<br />
Problem A.1. Prove that, if x is an element of Z/nZ, then x^2 is not a primitive root.<br />
<br />
Problem A.2. Prove that there are infinitely many primes p such that 2 is a square in Z/pZ. Hint: suppose there are only finitely many such primes p_1, .. p_r, and define N = (p_1 .. p_r)^2 - 2. Where can you go from here...?<br />
<br />
Problem A.3. Give a list of five primes p such that 2 is not a primitive root in Z/pZ (you can use the method of this proof or any other.)<br />
<br />
Problem B. Prove that 24 is the largest integer n such that every element of (Z/nZ)^* is a root of x^2-1.<br />
<br />
* '''Oct 19''': Problem A. Using p = 23 and q=31, show how to encrypt the message x=240 with Rabin's algorithm. Find all possible decryptions of your encrypted message.<br />
<br />
Problem B. Give a prime factorization of the Gaussian integer 7+9i.<br />
<br />
Problem C. Read the notes from [http://www.math.ucsd.edu/~jverstra/Gaussian1.pdf here]. Note that Z[i] satisfies a ''reduction theorem'': if n and d are Gaussian integers, then there exists integers q and r such that n = qd + r and Norm(r) < Norm(d). But (by contrast with the case of Z) this d may not be unique. In some contexts it is better to be able to choose r uniquely, even if this means letting r have norm greater than Norm(d). <br />
<br />
C.1. When d = 1+2i, show that, for each n in Z[i], there is a '''unique''' pair (q,r) in Z[i] such that n = qd+r and r is contained in the set {0,1,2,3,4}. For instance, i can be written as i(1+2i) + 2, so we say i reduces to 2 mod (1+2i).<br />
(Hint: Suppose q(1+2i)+r = q'(1+2i) + r'. What can we say about (r-r'), and why is this incompatible with both r and r' being in {0,1,2,3,4}?<br />
<br />
C.2. Show that if n is an integer in Z, the reduction of n mod (1+2i) is equal to its reduction mod 5.<br />
<br />
Problem D. Let's try to figure out how to define "phi(d)" for a Gaussian integer d. Suppose S is a set of Gaussian integers such that every n in Z[i] can be written uniquely as qd+r, with q in Z[i] and r in S. (So for instance when d=1+2i, we showed in problem B that we can take S to be {0...4}. It would also be OK to take S to be {1..5} or {0,i,2i,1+i,1+2i}. In fact, it turns out that S has to be a set of size Norm(d) (I might or might not prove this in class; if not, feel free just to accept it.)<br />
<br />
Now define phi(d) to be the number of elements s of S such that s and d are coprime.<br />
<br />
D.1. Compute phi(1+2i) and phi(3).<br />
<br />
D.2. Prove that the value of phi(d) does not depend on the choice of S.<br />
<br />
D.3. Prove that every n in Z[i] which is coprime to 3 satisfies n^phi(3) = 1 mod 3; that is, Euler's theorem holds in this case. (You can prove this by direct computation; of course, if you want, you are welcome to prove that Euler's theorem holds for Z[i] in general, adapting the proof in Stein's book or the one we gave in class.)<br />
<br />
*'''Nov 2:''' 4.1 from the book.<br />
<br />
Problem A. Express 50005 as the sum of two squares.<br />
<br />
In the next two problems we denote by r(n) the number of ways to express n as the sum of two squares (i.e. the number of pairs (a,b) such that a^2 + b^2 = n.) For instance, r(5) = 8 (as shown on the midterm.)<br />
<br />
Problem B. Prove that, for any N, there exists an integer n such that r(n) > N. (I.E., the function r(n) is ''unbounded.''<br />
<br />
Problem C. If you like Sage, write a short program in Sage to compute r(n) and compute the average of r(n) as n ranges from 1 to 1000. Whether or not you like Sage, make a guess as to how this average would behave if you replaced "1000" by a larger and larger number. (Feel free to ask the Sage-lovers what answer they got in the optional first part of the question.) Can you prove this guess is correct?<br />
<br />
Problem D. We saw in class that the ring Z[sqrt(-5)] doesn't have unique factorization; 6 can be factored as 2*3 or (1+sqrt(-5))(1-sqrt(-5)). In this problem, we will prove that Z[sqrt(-d)] fails to have unique factorization for EVERY odd d >= 5. (Actually it's true for all d >= 5 but to make the proof manageable we'll restrict to the odd case.]<br />
<br />
D.1. Show that (1+sqrt(-d)), (1-sqrt(-d)) and 2 are irreducible in Z[sqrt(-d)].<br />
<br />
D.2. Now give an element of Z[sqrt(-d)] that has two distinct factorizations into irreducibles. (Hint: imitate the example we used for Z[sqrt(-5)].)<br />
''Remark:'' The rings Z[sqrt(d)], where d is positive, are quite different -- here we believe that there are infinitely many with unique factorization, though this conjecture has remained unproved for many decades!<br />
<br />
*'''November 16'''<br />
<br />
Book problems: 5.3,5.4,5.5<br />
<br />
Problem A: We discussed in class the problem of studying which positive integers are the sum of two squares. In this problem we prove that every element n of (Z/pZ) is the sum of two squares. We argue as follows. Let S be the set of squares in (Z/pZ), and let T be the set of (Z/pZ) consisting of all elements of the form n - x^2, for some x in (Z/pZ).<br />
<br />
A.1. What is the size of S and of T? Use this to show that S and T are not disjoint.<br />
<br />
A.2. Given that S and T are not disjoint, prove that n is the sum of two squares.<br />
<br />
Problem B. Using the continued fraction expansion, find a solution to the Pell equation x^2 - 13 y^2 = 1.<br />
<br />
Problem C. Show that the modified Pell equation x^2 - 7y^2 = -1 has no solutions in integers x,y. (Hint: reduce the equation modulo a suitably chosen prime.)<br />
<br />
Problem D. Stuffy Stirnweiss finished the 1945 season with a batting average of .3085443. Using continued fractions, guess how many at-bats he had. Tony Cuccinello had a batting average of .3084577. Given that he had more than 200 and fewer than 600 at-bats, can you estimate the number of at-bats he had?<br />
<br />
'''December 7'''<br />
<br />
Problem A. When p is a prime congruent to 3 mod 4, prove that ((p-1)/2)! is either 1 or -1 in (Z/pZ). '''OPTIONAL:''' Use sage to compute whether this factorial is 1 or -1 for many primes p. Is there any pattern? Does it seem to be 1 half the time and -1 half the time? Note that I have no idea what the answer to this question is.<br />
<br />
Problem B. Using the continued fraction expansion, find a solution to the Pell equation x^2 - 13 y^2 = 1.<br />
<br />
Problem C. Show that the modified Pell equation x^2 - 7y^2 = -1 has no solutions in integers x,y. (Hint: reduce the equation modulo a suitably chosen prime.)<br />
<br />
Problem D.1. Pick two values of a in F_11 = Z/11Z (a not equal to 3), such that the equation y^2 = x^3+ax+1 defines an elliptic curve (i.e., it is smooth). For each such a, determine the number of points #E(F_11) and check that it falls inside the interval described in class. <br />
<br />
Problem D.2. The point P = (0,1) lies on each of these curves. For a =3 (the curve discussed in class) determine the order of P in the elliptic curve group, that is, find the smallest positive integer n such that nP = (infinity) -- the identity element in the group law.<br />
<br />
<!-- '''November 19'''<br />
<br />
Problem A. Using the method discussed in class (which is also the method of problem 4.6 in Stein, which in retrospect I think was too hard to assign with no preparation) find a nontrivial solution with y > 0 to the equation x^2 + 11y^2 = z^2. <br />
Problem B. Let f(X) be the number of solutions of A^2 + B^2 = C^3 such that A^2, B^2, and C^3 are all at most X. Use the heuristic described in class to explain why one might expect f(X) to grow more or less like X^{1/3}.<br />
<br />
Problem C. Show that if A+Bi is the cube of a Gaussian integer, then A^2 + B^2 is a perfect cube.<br />
<br />
Do ONE of problem D1 and D2; D1 is for people who like Sage, D2 is for people who like proving things.<br />
<br />
D1. Use Sage to compute f(X) for X = 1000, 10000, 100000. Does the answer look consistent with the heuristic prediction you made in B? Does f(X)/X^{1/3} appear to be approaching a limit?<br />
D2. We will use problem C to give a lower bound for f(X). Use this to show that there are at least X^{1/3} Gaussian integers with norm at most X that are perfect cubes. From here, show that f(X) > X^{1/3}.<br />
<br />
OPTIONAL: Show that the converse of problem C also holds -- A+Bi has norm a perfect cube if and only if A+Bi is a perfect cube. (You will need unique factorization of Gaussian integers.) Using this fact, prove that f(X) is bounded above by C X^{1/3} for some constant C.<br />
<br />
'''December 8 (note nonstandard due date)'''<br />
<br />
Problem A: Hyperbolas, ellipses, and "magic slopes"<br />
<br />
In the usual analytic geometry, both hyperbolas and ellipses are given by equations of the form<br />
<br />
ax^2 + bxy + cy^2 = d (*)<br />
<br />
<br />
(we always assume d is nonzero.)<br />
<br />
How can we tell whether such an equation describes an ellipse or a<br />
hyperbola? Well, in the geometric setting we all know and love (i.e. in<br />
R^2) a hyperbola has asymptotes and an ellipse does not. What is an <br />
asymptote? You could say it's "a line which doesn't intersect the curve", <br />
but ellipses have such lines too. You might want to say "a line which <br />
doesn't intersect the curve but comes closer and closer to the curve," but <br />
the problem here is that this a) is somewhat imprecise, and b) relies on a <br />
notion of "closer" that is not going to be very clear in Z/pZ, which is <br />
our ultimate goal!<br />
<br />
<br />
So let me state it a different way: let's say the asymptotes of a <br />
hyperbola have slopes m_1 and m_2. These are "magic slopes" in the <br />
following sense: ANY line of slope m_1 (and ditto for m_2) intersects the <br />
hyperbola in at most one point. An ellipse doesn't have "magic slopes" -- <br />
you can convince yourself by drawing pictures that, for any slope m, you <br />
can find a line of slope m striking the ellipse twice.<br />
<br />
<br />
[REMARK: For technical reasons we're going to ignore hyperbolae with a <br />
vertical asymptote, since vertical lines don't exactly have slope.]<br />
<br />
<br />
Now you might know the criterion that, over the real numbers.<br />
<br />
<br />
ax^2 + bxy + cy^2 = d<br />
<br />
<br />
is a hyperbola if b^2 - 4ac > 0 and an ellipse if b^2 - 4ac < 0. (For <br />
instance, x^2 - y^2 = 1 is a hyperbola with magic slopes 1 and -1, and x^2 <br />
+ y^2 = 1 is an ellipse.) <br />
This criterion does not make sense in Z/pZ, where there is no notion of <br />
"greater" or "less."<br />
<br />
<br />
QUESTION A.1.<br />
<br />
For which primes p is it the case that <br />
<br />
x^2 + y^2 = 1<br />
<br />
has magic slopes in Z/pZ? <br />
<br />
QUESTION A.2. For which primes p is it the case that<br />
<br />
<br />
x^2 - y^2 = 1<br />
<br />
<br />
has magic slopes in Z/pZ?<br />
<br />
<br />
QUESTION B: Back in the very first week of this course we studied the set of integers x such that x^2 + 1 is prime. Based on the heuristics we discussed in class, about how many x between 1 and N would you expect to satisfy "x^2 + 1 is prime?"<br />
<br />
<br />
QUESTION C: Let z be a complex number. Then the set of all complex numbers of the form a + bz, with a and b integers, forms a lattice in the complex plane (thought of as R^2.) What is the covolume of this lattice, in terms of z?<br />
<br />
<br />
QUESTION D: Let z be a complex number. Using Minkowski's theorem and the result of question C, show that there exist integers a,b, not both zero, such that Norm(a+bz) is at most (4/pi) Im(z). (Hint: let Omega be the set of all complex numbers of norm at most (4/pi) Im(z).)<br />
<br />
<br />
OPTIONAL EXTRA: Can you give an exact formula for the minimal norm of any nonzero complex number of the form a+bz?<br />
<br />
'''December 12''' Book problems 6.1, 6.2, 6.5, 6.10. For extra credit attempt to do problem 6.9.<br />
<br />
--></div>Andreichttps://www.math.wisc.edu/wiki/index.php?title=Math_567_--_Elementary_Number_Theory&diff=18694Math 567 -- Elementary Number Theory2020-01-19T16:58:30Z<p>Andreic: </p>
<hr />
<div>'''MATH 567<br />
<br />
Elementary Number Theory'''<br />
<br />
MWF 1:20-2:10, Van Vleck B119<br />
<br />
'''Professor:''' [http://www.math.wisc.edu/~andreic/ Andrei Caldararu] (andreic@math.wisc.edu)<br />
''Office Hours:'' Wednesdays 3:30-4:30, Van Vleck 605. <br />
<br />
'''Grader:''' TBA<br />
''Office Hours:'' TBA.<br />
<br />
Math 567 is a course in elementary number theory, aimed at undergraduates majoring in math or other quantitative disciplines. A general familiarity with abstract algebra at the level of Math 541 will be assumed, but students who haven't taken 541 are welcome to attend if they're willing to play a little catchup. We will be using William Stein's new (and cheap) textbook [http://www.amazon.com/Elementary-Number-Theory-Computational-Undergraduate/dp/0387855246 Elementary Number Theory: Primes, Congruences, and Secrets], which emphasizes computational approaches to the subject. If you don't need a physical copy of the book, [http://www.williamstein.org/ent/ it is available as a free legal .pdf.] We will be using the (free, public-domain) mathematical software [http://www.sagemath.org/ SAGE], developed largely by Stein, as an integral component of our coursework. There is a [http://doc.sagemath.org/pdf/en/tutorial/SageTutorial.pdf useful online tutorial.] You can download SAGE to your own computer or [http://www.sagenb.org use it online].<br />
<br />
Topics include some subset of, but are not limited to: Divisibility, the Euclidean algorithm and the GCD, linear Diophantine equations, prime numbers and uniqueness of factorization. Congruences, Chinese remainder theorem, Fermat's "little" theorem, Wilson's theorem, Euler's theorem and totient function, the RSA cryptosystem, Rabin's encryption scheme, Diffie-Hellman key exchange protocol. Number-theoretic functions, multiplicative functions, Möbius inversion. Primitive roots and indices. Quadratic reciprocity and the Legendre symbol. Perfect numbers, Mersenne primes, Fermat primes. Pythagorean triples, Special cases of Fermat's "last" theorem. Fibonacci numbers. Continued fractions. Distribution of primes, discussion of prime number theorem. Primality testing and factoring algorithms.<br />
<br />
'''Course Policies:''' Homework will be due on Fridays. It can be turned in late only with ''advance'' permission from your grader. It is acceptable to use calculators and computers on homework (indeed, some of it will require a computer) but calculators are not allowed during exams. You are encouraged to work together on homework, but writeups must be done individually.<br />
<br />
Many of the problems in this course will ask you to prove things. I expect proofs to be written in English sentences; the proofs in Stein's book are a good model for the level of verbosity I am looking for. <br />
<br />
'''Grading:''' The grade in Math 567 will be composed of 50% homework, 25% midterm, 25% final. The midterm will be on March 11, in class. The final exam will be on 5/8/2020, 7:45AM-9:45AM in room TBA.<br />
<br />
'''Syllabus:''' <br />
(This may change as we see what pace works well for the course. All section numbers refer to Stein's book.)<br />
<br />
* Sep 7 + Sep 11-15: Prime numbers, prime factorizations, Euclidean algorithm and GCD (1.1-1.2)<br />
* Sep 18-22: The integers mod n, Euler's theorem, the phi function (2.1-2.2)<br />
* Sep 25-29: Modular exponentiation, primality testing, and primitive roots (2.4-2.5)<br />
* Oct 2-6: Public-key cryptography and RSA (3.1-3.4)<br />
* Oct 9-13: Rabin's algorithm (not in the book); algebraic numbers <br />
* Oct 16-20: Quadratic reciprocity (4.1-4.4)<br />
* Oct 23-27: Finite and infinite continued fractions (5.1-5.3)<br />
* Oct 31, Nov 2, Nov 7: Continued fractions and diophantine approximation (5.4-5.5)<br />
* Nov. 9: ''Midterm exam''<br />
* Nov 13-17: Diophantine equations I: Pell's equation and Lagrange's theorem<br />
* Nov 21 and Nov 28-30: Elliptic curves (6.1-6.2)<br />
* Dec. 4-8: Applications of elliptic curves (6.3-6.4)<br />
* Dec. 12: advanced topic TBD: maybe additional discussion of cryptographic techniques?<br />
<br />
'''Homework:'''<br />
Homework is due at the beginning of class on the specified Friday. Typing your homework is not a requirement, but if you don't already know LaTeX I highly recommend that you learn it and use it to typeset your homework. I will sometimes assign extra problems, which I will e-mail to the class list and include here.<br />
<br />
* '''Jan 31''': 1.1, 1.3, 1.5, 1.7 (use SAGE), 1.8, 1.14.<br />
Problem A: Use SAGE to compute the number of x in [1..N] such that x^2 + 1 is prime, for N = 100, N = 1000, and N = 10000. Let f(N) be the number of such x. <br />
<br />
a) Can you formulate a conjecture about the relationship between f(N) and N/log N? <br />
<br />
b) What if x^2 + 1 is replaced with x^2 + 2? Can you explain why x^2 + 2 appears less likely to be prime? (Hint: consider x mod 3.)<br />
<br />
c) Prove that f(N) is at most (1/2)N+1. (Hint: consider x mod 2.) <br />
<br />
d) Give as good an upper bound as you can for f(N).<br />
<br />
Note that, despite the evident regularities you'll observe in this problem, we do not even know whether there are infinitely many primes of the form x^2 + 1! You would become very famous if you proved this.<br />
<br />
<!--<br />
* '''Sep 28''': 2.6 (the formulation of numerical evidence should be done by Sage if you've got Sage working, and by calculator if not; you can use an online tool like [http://primes.utm.edu/curios/includes/primetest.php this] to test whether a number is prime.) 2.8,2.9,2.11,2.12,2.14,2.19<br />
<br />
* '''Oct 5''': 2.15, 2.18, 2.23, 2.26, 2.30.<br />
<br />
Problem A: Prove that if n=pq, with p,q prime, then n is not a Carmichael number.<br />
<br />
* '''Oct 12''': Book problems: 3.4, 3.5, 3.6<br />
<br />
Problem A. Prove that there are infinitely many primes p such that 2 is '''not''' a primitive root in Z/pZ. We break this up into steps.<br />
<br />
Problem A.1. Prove that, if x is an element of Z/nZ, then x^2 is not a primitive root.<br />
<br />
Problem A.2. Prove that there are infinitely many primes p such that 2 is a square in Z/pZ. Hint: suppose there are only finitely many such primes p_1, .. p_r, and define N = (p_1 .. p_r)^2 - 2. Where can you go from here...?<br />
<br />
Problem A.3. Give a list of five primes p such that 2 is not a primitive root in Z/pZ (you can use the method of this proof or any other.)<br />
<br />
Problem B. Prove that 24 is the largest integer n such that every element of (Z/nZ)^* is a root of x^2-1.<br />
<br />
* '''Oct 19''': Problem A. Using p = 23 and q=31, show how to encrypt the message x=240 with Rabin's algorithm. Find all possible decryptions of your encrypted message.<br />
<br />
Problem B. Give a prime factorization of the Gaussian integer 7+9i.<br />
<br />
Problem C. Read the notes from [http://www.math.ucsd.edu/~jverstra/Gaussian1.pdf here]. Note that Z[i] satisfies a ''reduction theorem'': if n and d are Gaussian integers, then there exists integers q and r such that n = qd + r and Norm(r) < Norm(d). But (by contrast with the case of Z) this d may not be unique. In some contexts it is better to be able to choose r uniquely, even if this means letting r have norm greater than Norm(d). <br />
<br />
C.1. When d = 1+2i, show that, for each n in Z[i], there is a '''unique''' pair (q,r) in Z[i] such that n = qd+r and r is contained in the set {0,1,2,3,4}. For instance, i can be written as i(1+2i) + 2, so we say i reduces to 2 mod (1+2i).<br />
(Hint: Suppose q(1+2i)+r = q'(1+2i) + r'. What can we say about (r-r'), and why is this incompatible with both r and r' being in {0,1,2,3,4}?<br />
<br />
C.2. Show that if n is an integer in Z, the reduction of n mod (1+2i) is equal to its reduction mod 5.<br />
<br />
Problem D. Let's try to figure out how to define "phi(d)" for a Gaussian integer d. Suppose S is a set of Gaussian integers such that every n in Z[i] can be written uniquely as qd+r, with q in Z[i] and r in S. (So for instance when d=1+2i, we showed in problem B that we can take S to be {0...4}. It would also be OK to take S to be {1..5} or {0,i,2i,1+i,1+2i}. In fact, it turns out that S has to be a set of size Norm(d) (I might or might not prove this in class; if not, feel free just to accept it.)<br />
<br />
Now define phi(d) to be the number of elements s of S such that s and d are coprime.<br />
<br />
D.1. Compute phi(1+2i) and phi(3).<br />
<br />
D.2. Prove that the value of phi(d) does not depend on the choice of S.<br />
<br />
D.3. Prove that every n in Z[i] which is coprime to 3 satisfies n^phi(3) = 1 mod 3; that is, Euler's theorem holds in this case. (You can prove this by direct computation; of course, if you want, you are welcome to prove that Euler's theorem holds for Z[i] in general, adapting the proof in Stein's book or the one we gave in class.)<br />
<br />
*'''Nov 2:''' 4.1 from the book.<br />
<br />
Problem A. Express 50005 as the sum of two squares.<br />
<br />
In the next two problems we denote by r(n) the number of ways to express n as the sum of two squares (i.e. the number of pairs (a,b) such that a^2 + b^2 = n.) For instance, r(5) = 8 (as shown on the midterm.)<br />
<br />
Problem B. Prove that, for any N, there exists an integer n such that r(n) > N. (I.E., the function r(n) is ''unbounded.''<br />
<br />
Problem C. If you like Sage, write a short program in Sage to compute r(n) and compute the average of r(n) as n ranges from 1 to 1000. Whether or not you like Sage, make a guess as to how this average would behave if you replaced "1000" by a larger and larger number. (Feel free to ask the Sage-lovers what answer they got in the optional first part of the question.) Can you prove this guess is correct?<br />
<br />
Problem D. We saw in class that the ring Z[sqrt(-5)] doesn't have unique factorization; 6 can be factored as 2*3 or (1+sqrt(-5))(1-sqrt(-5)). In this problem, we will prove that Z[sqrt(-d)] fails to have unique factorization for EVERY odd d >= 5. (Actually it's true for all d >= 5 but to make the proof manageable we'll restrict to the odd case.]<br />
<br />
D.1. Show that (1+sqrt(-d)), (1-sqrt(-d)) and 2 are irreducible in Z[sqrt(-d)].<br />
<br />
D.2. Now give an element of Z[sqrt(-d)] that has two distinct factorizations into irreducibles. (Hint: imitate the example we used for Z[sqrt(-5)].)<br />
''Remark:'' The rings Z[sqrt(d)], where d is positive, are quite different -- here we believe that there are infinitely many with unique factorization, though this conjecture has remained unproved for many decades!<br />
<br />
*'''November 16'''<br />
<br />
Book problems: 5.3,5.4,5.5<br />
<br />
Problem A: We discussed in class the problem of studying which positive integers are the sum of two squares. In this problem we prove that every element n of (Z/pZ) is the sum of two squares. We argue as follows. Let S be the set of squares in (Z/pZ), and let T be the set of (Z/pZ) consisting of all elements of the form n - x^2, for some x in (Z/pZ).<br />
<br />
A.1. What is the size of S and of T? Use this to show that S and T are not disjoint.<br />
<br />
A.2. Given that S and T are not disjoint, prove that n is the sum of two squares.<br />
<br />
Problem B. Using the continued fraction expansion, find a solution to the Pell equation x^2 - 13 y^2 = 1.<br />
<br />
Problem C. Show that the modified Pell equation x^2 - 7y^2 = -1 has no solutions in integers x,y. (Hint: reduce the equation modulo a suitably chosen prime.)<br />
<br />
Problem D. Stuffy Stirnweiss finished the 1945 season with a batting average of .3085443. Using continued fractions, guess how many at-bats he had. Tony Cuccinello had a batting average of .3084577. Given that he had more than 200 and fewer than 600 at-bats, can you estimate the number of at-bats he had?<br />
<br />
'''December 7'''<br />
<br />
Problem A. When p is a prime congruent to 3 mod 4, prove that ((p-1)/2)! is either 1 or -1 in (Z/pZ). '''OPTIONAL:''' Use sage to compute whether this factorial is 1 or -1 for many primes p. Is there any pattern? Does it seem to be 1 half the time and -1 half the time? Note that I have no idea what the answer to this question is.<br />
<br />
Problem B. Using the continued fraction expansion, find a solution to the Pell equation x^2 - 13 y^2 = 1.<br />
<br />
Problem C. Show that the modified Pell equation x^2 - 7y^2 = -1 has no solutions in integers x,y. (Hint: reduce the equation modulo a suitably chosen prime.)<br />
<br />
Problem D.1. Pick two values of a in F_11 = Z/11Z (a not equal to 3), such that the equation y^2 = x^3+ax+1 defines an elliptic curve (i.e., it is smooth). For each such a, determine the number of points #E(F_11) and check that it falls inside the interval described in class. <br />
<br />
Problem D.2. The point P = (0,1) lies on each of these curves. For a =3 (the curve discussed in class) determine the order of P in the elliptic curve group, that is, find the smallest positive integer n such that nP = (infinity) -- the identity element in the group law.<br />
<br />
<!-- '''November 19'''<br />
<br />
Problem A. Using the method discussed in class (which is also the method of problem 4.6 in Stein, which in retrospect I think was too hard to assign with no preparation) find a nontrivial solution with y > 0 to the equation x^2 + 11y^2 = z^2. <br />
Problem B. Let f(X) be the number of solutions of A^2 + B^2 = C^3 such that A^2, B^2, and C^3 are all at most X. Use the heuristic described in class to explain why one might expect f(X) to grow more or less like X^{1/3}.<br />
<br />
Problem C. Show that if A+Bi is the cube of a Gaussian integer, then A^2 + B^2 is a perfect cube.<br />
<br />
Do ONE of problem D1 and D2; D1 is for people who like Sage, D2 is for people who like proving things.<br />
<br />
D1. Use Sage to compute f(X) for X = 1000, 10000, 100000. Does the answer look consistent with the heuristic prediction you made in B? Does f(X)/X^{1/3} appear to be approaching a limit?<br />
D2. We will use problem C to give a lower bound for f(X). Use this to show that there are at least X^{1/3} Gaussian integers with norm at most X that are perfect cubes. From here, show that f(X) > X^{1/3}.<br />
<br />
OPTIONAL: Show that the converse of problem C also holds -- A+Bi has norm a perfect cube if and only if A+Bi is a perfect cube. (You will need unique factorization of Gaussian integers.) Using this fact, prove that f(X) is bounded above by C X^{1/3} for some constant C.<br />
<br />
'''December 8 (note nonstandard due date)'''<br />
<br />
Problem A: Hyperbolas, ellipses, and "magic slopes"<br />
<br />
In the usual analytic geometry, both hyperbolas and ellipses are given by equations of the form<br />
<br />
ax^2 + bxy + cy^2 = d (*)<br />
<br />
<br />
(we always assume d is nonzero.)<br />
<br />
How can we tell whether such an equation describes an ellipse or a<br />
hyperbola? Well, in the geometric setting we all know and love (i.e. in<br />
R^2) a hyperbola has asymptotes and an ellipse does not. What is an <br />
asymptote? You could say it's "a line which doesn't intersect the curve", <br />
but ellipses have such lines too. You might want to say "a line which <br />
doesn't intersect the curve but comes closer and closer to the curve," but <br />
the problem here is that this a) is somewhat imprecise, and b) relies on a <br />
notion of "closer" that is not going to be very clear in Z/pZ, which is <br />
our ultimate goal!<br />
<br />
<br />
So let me state it a different way: let's say the asymptotes of a <br />
hyperbola have slopes m_1 and m_2. These are "magic slopes" in the <br />
following sense: ANY line of slope m_1 (and ditto for m_2) intersects the <br />
hyperbola in at most one point. An ellipse doesn't have "magic slopes" -- <br />
you can convince yourself by drawing pictures that, for any slope m, you <br />
can find a line of slope m striking the ellipse twice.<br />
<br />
<br />
[REMARK: For technical reasons we're going to ignore hyperbolae with a <br />
vertical asymptote, since vertical lines don't exactly have slope.]<br />
<br />
<br />
Now you might know the criterion that, over the real numbers.<br />
<br />
<br />
ax^2 + bxy + cy^2 = d<br />
<br />
<br />
is a hyperbola if b^2 - 4ac > 0 and an ellipse if b^2 - 4ac < 0. (For <br />
instance, x^2 - y^2 = 1 is a hyperbola with magic slopes 1 and -1, and x^2 <br />
+ y^2 = 1 is an ellipse.) <br />
This criterion does not make sense in Z/pZ, where there is no notion of <br />
"greater" or "less."<br />
<br />
<br />
QUESTION A.1.<br />
<br />
For which primes p is it the case that <br />
<br />
x^2 + y^2 = 1<br />
<br />
has magic slopes in Z/pZ? <br />
<br />
QUESTION A.2. For which primes p is it the case that<br />
<br />
<br />
x^2 - y^2 = 1<br />
<br />
<br />
has magic slopes in Z/pZ?<br />
<br />
<br />
QUESTION B: Back in the very first week of this course we studied the set of integers x such that x^2 + 1 is prime. Based on the heuristics we discussed in class, about how many x between 1 and N would you expect to satisfy "x^2 + 1 is prime?"<br />
<br />
<br />
QUESTION C: Let z be a complex number. Then the set of all complex numbers of the form a + bz, with a and b integers, forms a lattice in the complex plane (thought of as R^2.) What is the covolume of this lattice, in terms of z?<br />
<br />
<br />
QUESTION D: Let z be a complex number. Using Minkowski's theorem and the result of question C, show that there exist integers a,b, not both zero, such that Norm(a+bz) is at most (4/pi) Im(z). (Hint: let Omega be the set of all complex numbers of norm at most (4/pi) Im(z).)<br />
<br />
<br />
OPTIONAL EXTRA: Can you give an exact formula for the minimal norm of any nonzero complex number of the form a+bz?<br />
<br />
'''December 12''' Book problems 6.1, 6.2, 6.5, 6.10. For extra credit attempt to do problem 6.9.<br />
<br />
--></div>Andreichttps://www.math.wisc.edu/wiki/index.php?title=Math_567_--_Elementary_Number_Theory&diff=18693Math 567 -- Elementary Number Theory2020-01-19T16:57:44Z<p>Andreic: </p>
<hr />
<div>'''MATH 567<br />
<br />
Elementary Number Theory'''<br />
<br />
MWF 1:20-2:10, Van Vleck B119<br />
<br />
'''Professor:''' [http://www.math.wisc.edu/~andreic/ Andrei Caldararu] (andreic@math.wisc.edu)<br />
''Office Hours:'' Wednesdays 3:30-4:30, Van Vleck 605. <br />
<br />
'''Grader:''' TBA<br />
''Office Hours:'' TBA.<br />
<br />
Math 567 is a course in elementary number theory, aimed at undergraduates majoring in math or other quantitative disciplines. A general familiarity with abstract algebra at the level of Math 541 will be assumed, but students who haven't taken 541 are welcome to attend if they're willing to play a little catchup. We will be using William Stein's new (and cheap) textbook [http://www.amazon.com/Elementary-Number-Theory-Computational-Undergraduate/dp/0387855246 Elementary Number Theory: Primes, Congruences, and Secrets], which emphasizes computational approaches to the subject. If you don't need a physical copy of the book, [http://www.williamstein.org/ent/ it is available as a free legal .pdf.] We will be using the (free, public-domain) mathematical software [http://www.sagemath.org/ SAGE], developed largely by Stein, as an integral component of our coursework. There is a [http://doc.sagemath.org/pdf/en/tutorial/SageTutorial.pdf useful online tutorial.] You can download SAGE to your own computer or [http://www.sagenb.org use it online].<br />
<br />
Topics include some subset of, but are not limited to: Divisibility, the Euclidean algorithm and the GCD, linear Diophantine equations, prime numbers and uniqueness of factorization. Congruences, Chinese remainder theorem, Fermat's "little" theorem, Wilson's theorem, Euler's theorem and totient function, the RSA cryptosystem, Rabin's encryption scheme, Diffie-Hellman key exchange protocol. Number-theoretic functions, multiplicative functions, Möbius inversion. Primitive roots and indices. Quadratic reciprocity and the Legendre symbol. Perfect numbers, Mersenne primes, Fermat primes. Pythagorean triples, Special cases of Fermat's "last" theorem. Fibonacci numbers. Continued fractions. Distribution of primes, discussion of prime number theorem. Primality testing and factoring algorithms.<br />
<br />
'''Course Policies:''' Homework will be due on Fridays. It can be turned in late only with ''advance'' permission from your grader. It is acceptable to use calculators and computers on homework (indeed, some of it will require a computer) but calculators are not allowed during exams. You are encouraged to work together on homework, but writeups must be done individually.<br />
<br />
Many of the problems in this course will ask you to prove things. I expect proofs to be written in English sentences; the proofs in Stein's book are a good model for the level of verbosity I am looking for. <br />
<br />
'''Grading:''' The grade in Math 567 will be composed of 50% homework, 25% midterm, 25% final. The midterm will be on March 11, in class. The final exam will be on 5/8/2020, 7:45AM-9:45AM in room TBA.<br />
<br />
'''Syllabus:''' <br />
(This may change as we see what pace works well for the course. All section numbers refer to Stein's book.)<br />
<br />
* Sep 7 + Sep 11-15: Prime numbers, prime factorizations, Euclidean algorithm and GCD (1.1-1.2)<br />
* Sep 18-22: The integers mod n, Euler's theorem, the phi function (2.1-2.2)<br />
* Sep 25-29: Modular exponentiation, primality testing, and primitive roots (2.4-2.5)<br />
* Oct 2-6: Public-key cryptography and RSA (3.1-3.4)<br />
* Oct 9-13: Rabin's algorithm (not in the book); algebraic numbers <br />
* Oct 16-20: Quadratic reciprocity (4.1-4.4)<br />
* Oct 23-27: Finite and infinite continued fractions (5.1-5.3)<br />
* Oct 31, Nov 2, Nov 7: Continued fractions and diophantine approximation (5.4-5.5)<br />
* Nov. 9: ''Midterm exam''<br />
* Nov 13-17: Diophantine equations I: Pell's equation and Lagrange's theorem<br />
* Nov 21 and Nov 28-30: Elliptic curves (6.1-6.2)<br />
* Dec. 4-8: Applications of elliptic curves (6.3-6.4)<br />
* Dec. 12: advanced topic TBD: maybe additional discussion of cryptographic techniques?<br />
<br />
'''Homework:'''<br />
Homework is due at the beginning of class on the specified Friday. Typing your homework is not a requirement, but if you don't already know LaTeX I highly recommend that you learn it and use it to typeset your homework. I will sometimes assign extra problems, which I will e-mail to the class list and include here.<br />
<br />
* '''Jan 31''': 1.1, 1.3, 1.5, 1.7 (use SAGE), 1.8, 1.14.<br />
Problem A: Use SAGE to compute the number of x in [1..N] such that x^2 + 1 is prime, for N = 100, N = 1000, and N = 10000. Let f(N) be the number of such x. <br />
<br />
a) Can you formulate a conjecture about the relationship between f(N) and N/log N? <br />
<br />
b) What if x^2 + 1 is replaced with x^2 + 2? Can you explain why x^2 + 2 appears less likely to be prime? (Hint: consider x mod 3.)<br />
<br />
c) Prove that f(N) is at most (1/2)N+1. (Hint: consider x mod 2.) <br />
<br />
d) Give as good an upper bound as you can for f(N).<br />
<br />
Note that, despite the evident regularities you'll observe in this problem, we do not even know whether there are infinitely many primes of the form x^2 + 1! You would become very famous if you proved this.<br />
<br />
<!--<br />
* '''Sep 28''': 2.6 (the formulation of numerical evidence should be done by Sage if you've got Sage working, and by calculator if not; you can use an online tool like [http://primes.utm.edu/curios/includes/primetest.php this] to test whether a number is prime.) 2.8,2.9,2.11,2.12,2.14,2.19<br />
<br />
* '''Oct 5''': 2.15, 2.18, 2.23, 2.26, 2.30.<br />
<br />
Problem A: Prove that if n=pq, with p,q prime, then n is not a Carmichael number.<br />
<br />
* '''Oct 12''': Book problems: 3.4, 3.5, 3.6<br />
<br />
Problem A. Prove that there are infinitely many primes p such that 2 is '''not''' a primitive root in Z/pZ. We break this up into steps.<br />
<br />
Problem A.1. Prove that, if x is an element of Z/nZ, then x^2 is not a primitive root.<br />
<br />
Problem A.2. Prove that there are infinitely many primes p such that 2 is a square in Z/pZ. Hint: suppose there are only finitely many such primes p_1, .. p_r, and define N = (p_1 .. p_r)^2 - 2. Where can you go from here...?<br />
<br />
Problem A.3. Give a list of five primes p such that 2 is not a primitive root in Z/pZ (you can use the method of this proof or any other.)<br />
<br />
Problem B. Prove that 24 is the largest integer n such that every element of (Z/nZ)^* is a root of x^2-1.<br />
<br />
* '''Oct 19''': Problem A. Using p = 23 and q=31, show how to encrypt the message x=240 with Rabin's algorithm. Find all possible decryptions of your encrypted message.<br />
<br />
Problem B. Give a prime factorization of the Gaussian integer 7+9i.<br />
<br />
Problem C. Read the notes from [http://www.math.ucsd.edu/~jverstra/Gaussian1.pdf here]. Note that Z[i] satisfies a ''reduction theorem'': if n and d are Gaussian integers, then there exists integers q and r such that n = qd + r and Norm(r) < Norm(d). But (by contrast with the case of Z) this d may not be unique. In some contexts it is better to be able to choose r uniquely, even if this means letting r have norm greater than Norm(d). <br />
<br />
C.1. When d = 1+2i, show that, for each n in Z[i], there is a '''unique''' pair (q,r) in Z[i] such that n = qd+r and r is contained in the set {0,1,2,3,4}. For instance, i can be written as i(1+2i) + 2, so we say i reduces to 2 mod (1+2i).<br />
(Hint: Suppose q(1+2i)+r = q'(1+2i) + r'. What can we say about (r-r'), and why is this incompatible with both r and r' being in {0,1,2,3,4}?<br />
<br />
C.2. Show that if n is an integer in Z, the reduction of n mod (1+2i) is equal to its reduction mod 5.<br />
<br />
Problem D. Let's try to figure out how to define "phi(d)" for a Gaussian integer d. Suppose S is a set of Gaussian integers such that every n in Z[i] can be written uniquely as qd+r, with q in Z[i] and r in S. (So for instance when d=1+2i, we showed in problem B that we can take S to be {0...4}. It would also be OK to take S to be {1..5} or {0,i,2i,1+i,1+2i}. In fact, it turns out that S has to be a set of size Norm(d) (I might or might not prove this in class; if not, feel free just to accept it.)<br />
<br />
Now define phi(d) to be the number of elements s of S such that s and d are coprime.<br />
<br />
D.1. Compute phi(1+2i) and phi(3).<br />
<br />
D.2. Prove that the value of phi(d) does not depend on the choice of S.<br />
<br />
D.3. Prove that every n in Z[i] which is coprime to 3 satisfies n^phi(3) = 1 mod 3; that is, Euler's theorem holds in this case. (You can prove this by direct computation; of course, if you want, you are welcome to prove that Euler's theorem holds for Z[i] in general, adapting the proof in Stein's book or the one we gave in class.)<br />
<br />
*'''Nov 2:''' 4.1 from the book.<br />
<br />
Problem A. Express 50005 as the sum of two squares.<br />
<br />
In the next two problems we denote by r(n) the number of ways to express n as the sum of two squares (i.e. the number of pairs (a,b) such that a^2 + b^2 = n.) For instance, r(5) = 8 (as shown on the midterm.)<br />
<br />
Problem B. Prove that, for any N, there exists an integer n such that r(n) > N. (I.E., the function r(n) is ''unbounded.''<br />
<br />
Problem C. If you like Sage, write a short program in Sage to compute r(n) and compute the average of r(n) as n ranges from 1 to 1000. Whether or not you like Sage, make a guess as to how this average would behave if you replaced "1000" by a larger and larger number. (Feel free to ask the Sage-lovers what answer they got in the optional first part of the question.) Can you prove this guess is correct?<br />
<br />
Problem D. We saw in class that the ring Z[sqrt(-5)] doesn't have unique factorization; 6 can be factored as 2*3 or (1+sqrt(-5))(1-sqrt(-5)). In this problem, we will prove that Z[sqrt(-d)] fails to have unique factorization for EVERY odd d >= 5. (Actually it's true for all d >= 5 but to make the proof manageable we'll restrict to the odd case.]<br />
<br />
D.1. Show that (1+sqrt(-d)), (1-sqrt(-d)) and 2 are irreducible in Z[sqrt(-d)].<br />
<br />
D.2. Now give an element of Z[sqrt(-d)] that has two distinct factorizations into irreducibles. (Hint: imitate the example we used for Z[sqrt(-5)].)<br />
''Remark:'' The rings Z[sqrt(d)], where d is positive, are quite different -- here we believe that there are infinitely many with unique factorization, though this conjecture has remained unproved for many decades!<br />
<br />
*'''November 16'''<br />
<br />
Book problems: 5.3,5.4,5.5<br />
<br />
Problem A: We discussed in class the problem of studying which positive integers are the sum of two squares. In this problem we prove that every element n of (Z/pZ) is the sum of two squares. We argue as follows. Let S be the set of squares in (Z/pZ), and let T be the set of (Z/pZ) consisting of all elements of the form n - x^2, for some x in (Z/pZ).<br />
<br />
A.1. What is the size of S and of T? Use this to show that S and T are not disjoint.<br />
<br />
A.2. Given that S and T are not disjoint, prove that n is the sum of two squares.<br />
<br />
Problem B. Using the continued fraction expansion, find a solution to the Pell equation x^2 - 13 y^2 = 1.<br />
<br />
Problem C. Show that the modified Pell equation x^2 - 7y^2 = -1 has no solutions in integers x,y. (Hint: reduce the equation modulo a suitably chosen prime.)<br />
<br />
Problem D. Stuffy Stirnweiss finished the 1945 season with a batting average of .3085443. Using continued fractions, guess how many at-bats he had. Tony Cuccinello had a batting average of .3084577. Given that he had more than 200 and fewer than 600 at-bats, can you estimate the number of at-bats he had?<br />
<br />
'''December 7'''<br />
<br />
Problem A. When p is a prime congruent to 3 mod 4, prove that ((p-1)/2)! is either 1 or -1 in (Z/pZ). '''OPTIONAL:''' Use sage to compute whether this factorial is 1 or -1 for many primes p. Is there any pattern? Does it seem to be 1 half the time and -1 half the time? Note that I have no idea what the answer to this question is.<br />
<br />
Problem B. Using the continued fraction expansion, find a solution to the Pell equation x^2 - 13 y^2 = 1.<br />
<br />
Problem C. Show that the modified Pell equation x^2 - 7y^2 = -1 has no solutions in integers x,y. (Hint: reduce the equation modulo a suitably chosen prime.)<br />
<br />
Problem D.1. Pick two values of a in F_11 = Z/11Z (a not equal to 3), such that the equation y^2 = x^3+ax+1 defines an elliptic curve (i.e., it is smooth). For each such a, determine the number of points #E(F_11) and check that it falls inside the interval described in class. <br />
<br />
Problem D.2. The point P = (0,1) lies on each of these curves. For a =3 (the curve discussed in class) determine the order of P in the elliptic curve group, that is, find the smallest positive integer n such that nP = (infinity) -- the identity element in the group law.<br />
<br />
<!-- '''November 19'''<br />
<br />
Problem A. Using the method discussed in class (which is also the method of problem 4.6 in Stein, which in retrospect I think was too hard to assign with no preparation) find a nontrivial solution with y > 0 to the equation x^2 + 11y^2 = z^2. <br />
Problem B. Let f(X) be the number of solutions of A^2 + B^2 = C^3 such that A^2, B^2, and C^3 are all at most X. Use the heuristic described in class to explain why one might expect f(X) to grow more or less like X^{1/3}.<br />
<br />
Problem C. Show that if A+Bi is the cube of a Gaussian integer, then A^2 + B^2 is a perfect cube.<br />
<br />
Do ONE of problem D1 and D2; D1 is for people who like Sage, D2 is for people who like proving things.<br />
<br />
D1. Use Sage to compute f(X) for X = 1000, 10000, 100000. Does the answer look consistent with the heuristic prediction you made in B? Does f(X)/X^{1/3} appear to be approaching a limit?<br />
D2. We will use problem C to give a lower bound for f(X). Use this to show that there are at least X^{1/3} Gaussian integers with norm at most X that are perfect cubes. From here, show that f(X) > X^{1/3}.<br />
<br />
OPTIONAL: Show that the converse of problem C also holds -- A+Bi has norm a perfect cube if and only if A+Bi is a perfect cube. (You will need unique factorization of Gaussian integers.) Using this fact, prove that f(X) is bounded above by C X^{1/3} for some constant C.<br />
<br />
'''December 8 (note nonstandard due date)'''<br />
<br />
Problem A: Hyperbolas, ellipses, and "magic slopes"<br />
<br />
In the usual analytic geometry, both hyperbolas and ellipses are given by equations of the form<br />
<br />
ax^2 + bxy + cy^2 = d (*)<br />
<br />
<br />
(we always assume d is nonzero.)<br />
<br />
How can we tell whether such an equation describes an ellipse or a<br />
hyperbola? Well, in the geometric setting we all know and love (i.e. in<br />
R^2) a hyperbola has asymptotes and an ellipse does not. What is an <br />
asymptote? You could say it's "a line which doesn't intersect the curve", <br />
but ellipses have such lines too. You might want to say "a line which <br />
doesn't intersect the curve but comes closer and closer to the curve," but <br />
the problem here is that this a) is somewhat imprecise, and b) relies on a <br />
notion of "closer" that is not going to be very clear in Z/pZ, which is <br />
our ultimate goal!<br />
<br />
<br />
So let me state it a different way: let's say the asymptotes of a <br />
hyperbola have slopes m_1 and m_2. These are "magic slopes" in the <br />
following sense: ANY line of slope m_1 (and ditto for m_2) intersects the <br />
hyperbola in at most one point. An ellipse doesn't have "magic slopes" -- <br />
you can convince yourself by drawing pictures that, for any slope m, you <br />
can find a line of slope m striking the ellipse twice.<br />
<br />
<br />
[REMARK: For technical reasons we're going to ignore hyperbolae with a <br />
vertical asymptote, since vertical lines don't exactly have slope.]<br />
<br />
<br />
Now you might know the criterion that, over the real numbers.<br />
<br />
<br />
ax^2 + bxy + cy^2 = d<br />
<br />
<br />
is a hyperbola if b^2 - 4ac > 0 and an ellipse if b^2 - 4ac < 0. (For <br />
instance, x^2 - y^2 = 1 is a hyperbola with magic slopes 1 and -1, and x^2 <br />
+ y^2 = 1 is an ellipse.) <br />
This criterion does not make sense in Z/pZ, where there is no notion of <br />
"greater" or "less."<br />
<br />
<br />
QUESTION A.1.<br />
<br />
For which primes p is it the case that <br />
<br />
x^2 + y^2 = 1<br />
<br />
has magic slopes in Z/pZ? <br />
<br />
QUESTION A.2. For which primes p is it the case that<br />
<br />
<br />
x^2 - y^2 = 1<br />
<br />
<br />
has magic slopes in Z/pZ?<br />
<br />
<br />
QUESTION B: Back in the very first week of this course we studied the set of integers x such that x^2 + 1 is prime. Based on the heuristics we discussed in class, about how many x between 1 and N would you expect to satisfy "x^2 + 1 is prime?"<br />
<br />
<br />
QUESTION C: Let z be a complex number. Then the set of all complex numbers of the form a + bz, with a and b integers, forms a lattice in the complex plane (thought of as R^2.) What is the covolume of this lattice, in terms of z?<br />
<br />
<br />
QUESTION D: Let z be a complex number. Using Minkowski's theorem and the result of question C, show that there exist integers a,b, not both zero, such that Norm(a+bz) is at most (4/pi) Im(z). (Hint: let Omega be the set of all complex numbers of norm at most (4/pi) Im(z).)<br />
<br />
<br />
OPTIONAL EXTRA: Can you give an exact formula for the minimal norm of any nonzero complex number of the form a+bz? --><br />
<br />
'''December 12''' Book problems 6.1, 6.2, 6.5, 6.10. For extra credit attempt to do problem 6.9.<br />
<br />
--></div>Andreichttps://www.math.wisc.edu/wiki/index.php?title=Colloquia&diff=17523Colloquia2019-07-16T14:28:53Z<p>Andreic: /* Mathematics Colloquium */</p>
<hr />
<div>= Mathematics Colloquium =<br />
<br />
All colloquia are on Fridays at 4:00 pm in Van Vleck B239, '''unless otherwise indicated'''.<br />
<br />
<br />
<br />
==Fall 2019==<br />
{| cellpadding="8"<br />
!align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|Sept 6<br />
|<br />
|<br />
|-<br />
|Sept 13<br />
| [https://www.math.ksu.edu/~soibel/ Yan Soibelman] (Kansas State)<br />
|[[#Yan Soibelman (Kansas State)| Riemann-Hilbert correspondence and Fukaya categories ]]<br />
| Caldararu<br />
|<br />
|-<br />
|Sept 16 '''Monday Room 911'''<br />
| Alicia Dickenstein (Buenos Aires)<br />
|[[# TBA| TBA ]]<br />
| Craciun<br />
|<br />
|-<br />
|Sept 20<br />
| Jianfeng Lu (Duke)<br />
|[[#TBA | TBA]]<br />
| Qin<br />
|<br />
|-<br />
|Sept 27<br />
|Elchnanan Mossel (MIT) Distinguished Lecture<br />
|-<br />
|Oct 4<br />
| Matt Baker (Georgia Tech)<br />
|<br />
|-<br />
|Oct 11<br />
|<br />
|-<br />
|Oct 18<br />
|<br />
|-<br />
|Oct 25<br />
|<br />
|-<br />
|Nov 1<br />
|Possibly reserved for job talk?<br />
|<br />
|-<br />
|Nov 8<br />
|Reserved for job talk<br />
|<br />
|-<br />
|Nov 15<br />
|Reserved for job talk<br />
|<br />
|-<br />
|Nov 22<br />
|Reserved for job talk<br />
|<br />
|-<br />
|Nov 29<br />
|Thanksgiving<br />
|<br />
|-<br />
|Dec 6<br />
|Reserved for job talk<br />
|<br />
|-<br />
|Dec 13<br />
|Reserved for job talk<br />
|<br />
|}<br />
<br />
==Spring 2020==<br />
<br />
{| cellpadding="8"<br />
!align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|<br />
|-<br />
|Jan 24<br />
|<br />
|-<br />
|Jan 31<br />
|<br />
|-<br />
|Feb 7<br />
|<br />
|-<br />
|Feb 14<br />
|<br />
|-<br />
|Feb 21<br />
|<br />
|-<br />
|Feb 28<br />
|<br />
|-<br />
|March 6<br />
|<br />
|-<br />
|March 13<br />
|<br />
|-<br />
|March 20<br />
|Spring break<br />
|<br />
|-<br />
|March 27<br />
|<br />
|-<br />
|April 3<br />
|<br />
|-<br />
|April 10<br />
| Sarah Koch (Michigan)<br />
|<br />
| Bruce (WIMAW)<br />
|-<br />
|April 17<br />
|<br />
|-<br />
|April 24<br />
|<br />
|-<br />
|May 1<br />
|Robert Lazarsfeld (Stony Brook)<br />
|Distinguished lecture<br />
|Erman<br />
|}<br />
<br />
== Abstracts ==<br />
<br />
===Yan Soibelman (Kansas State)===<br />
<br />
Title: Riemann-Hilbert correspondence and Fukaya categories<br />
<br />
Abstract: In this talk I am going to discuss the role of Fukaya categories in the Riemann-Hilbert correspondence<br />
for differential, q-difference and elliptic difference equations in dimension one.<br />
This approach not only gives a unified answer for several versions of the Riemann-Hilbert correspondence but also leads to a natural formulation<br />
of the non-abelian Hodge theory in dimension one. It also explains why periodic monopoles<br />
should appear as harmonic objects in this generalized non-abelian Hodge theory.<br />
All that is a part of the bigger project ``Holomorphic Floer theory",<br />
joint with Maxim Kontsevich.<br />
<br />
== Past Colloquia ==<br />
<br />
[[Colloquia/Blank|Blank]]<br />
<br />
[[Colloquia/Spring2019|Spring 2019]]<br />
<br />
[[Colloquia/Fall2018|Fall 2018]]<br />
<br />
[[Colloquia/Spring2018|Spring 2018]]<br />
<br />
[[Colloquia/Fall2017|Fall 2017]]<br />
<br />
[[Colloquia/Spring2017|Spring 2017]]<br />
<br />
[[Archived Fall 2016 Colloquia|Fall 2016]]<br />
<br />
[[Colloquia/Spring2016|Spring 2016]]<br />
<br />
[[Colloquia/Fall2015|Fall 2015]]<br />
<br />
[[Colloquia/Spring2014|Spring 2015]]<br />
<br />
[[Colloquia/Fall2014|Fall 2014]]<br />
<br />
[[Colloquia/Spring2014|Spring 2014]]<br />
<br />
[[Colloquia/Fall2013|Fall 2013]]<br />
<br />
[[Colloquia 2012-2013|Spring 2013]]<br />
<br />
[[Colloquia 2012-2013#Fall 2012|Fall 2012]]</div>Andreichttps://www.math.wisc.edu/wiki/index.php?title=Colloquia&diff=17522Colloquia2019-07-16T14:27:23Z<p>Andreic: </p>
<hr />
<div>= Mathematics Colloquium =<br />
<br />
All colloquia are on Fridays at 4:00 pm in Van Vleck B239, '''unless otherwise indicated'''.<br />
<br />
<br />
<br />
==Fall 2019==<br />
{| cellpadding="8"<br />
!align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|Sept 6<br />
|<br />
|<br />
|-<br />
|Sept 13<br />
| [https://www.math.ksu.edu/~soibel/ Yan Soibelman] (Kansas State)<br />
|[[#Yan Soibelman (Kansas State)| Riemann-Hilbert correspondence and Fukaya categories. ]]<br />
| Caldararu<br />
|<br />
|-<br />
|Sept 16 '''Monday Room 911'''<br />
| Alicia Dickenstein (Buenos Aires)<br />
|[[# TBA| TBA ]]<br />
| Craciun<br />
|<br />
|-<br />
|Sept 20<br />
| Jianfeng Lu (Duke)<br />
|[[#TBA | TBA]]<br />
| Qin<br />
|<br />
|-<br />
|Sept 27<br />
|Elchnanan Mossel (MIT) Distinguished Lecture<br />
|-<br />
|Oct 4<br />
| Matt Baker (Georgia Tech)<br />
|<br />
|-<br />
|Oct 11<br />
|<br />
|-<br />
|Oct 18<br />
|<br />
|-<br />
|Oct 25<br />
|<br />
|-<br />
|Nov 1<br />
|Possibly reserved for job talk?<br />
|<br />
|-<br />
|Nov 8<br />
|Reserved for job talk<br />
|<br />
|-<br />
|Nov 15<br />
|Reserved for job talk<br />
|<br />
|-<br />
|Nov 22<br />
|Reserved for job talk<br />
|<br />
|-<br />
|Nov 29<br />
|Thanksgiving<br />
|<br />
|-<br />
|Dec 6<br />
|Reserved for job talk<br />
|<br />
|-<br />
|Dec 13<br />
|Reserved for job talk<br />
|<br />
|}<br />
<br />
==Spring 2020==<br />
<br />
{| cellpadding="8"<br />
!align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|<br />
|-<br />
|Jan 24<br />
|<br />
|-<br />
|Jan 31<br />
|<br />
|-<br />
|Feb 7<br />
|<br />
|-<br />
|Feb 14<br />
|<br />
|-<br />
|Feb 21<br />
|<br />
|-<br />
|Feb 28<br />
|<br />
|-<br />
|March 6<br />
|<br />
|-<br />
|March 13<br />
|<br />
|-<br />
|March 20<br />
|Spring break<br />
|<br />
|-<br />
|March 27<br />
|<br />
|-<br />
|April 3<br />
|<br />
|-<br />
|April 10<br />
| Sarah Koch (Michigan)<br />
|<br />
| Bruce (WIMAW)<br />
|-<br />
|April 17<br />
|<br />
|-<br />
|April 24<br />
|<br />
|-<br />
|May 1<br />
|Robert Lazarsfeld (Stony Brook)<br />
|Distinguished lecture<br />
|Erman<br />
|}<br />
<br />
== Abstracts ==<br />
<br />
===Yan Soibelman (Kansas State)===<br />
<br />
Title: Riemann-Hilbert correspondence and Fukaya categories<br />
<br />
Abstract: In this talk I am going to discuss the role of Fukaya categories in the Riemann-Hilbert correspondence<br />
for differential, q-difference and elliptic difference equations in dimension one.<br />
This approach not only gives a unified answer for several versions of the Riemann-Hilbert correspondence but also leads to a natural formulation<br />
of the non-abelian Hodge theory in dimension one. It also explains why periodic monopoles<br />
should appear as harmonic objects in this generalized non-abelian Hodge theory.<br />
All that is a part of the bigger project ``Holomorphic Floer theory",<br />
joint with Maxim Kontsevich.<br />
<br />
== Past Colloquia ==<br />
<br />
[[Colloquia/Blank|Blank]]<br />
<br />
[[Colloquia/Spring2019|Spring 2019]]<br />
<br />
[[Colloquia/Fall2018|Fall 2018]]<br />
<br />
[[Colloquia/Spring2018|Spring 2018]]<br />
<br />
[[Colloquia/Fall2017|Fall 2017]]<br />
<br />
[[Colloquia/Spring2017|Spring 2017]]<br />
<br />
[[Archived Fall 2016 Colloquia|Fall 2016]]<br />
<br />
[[Colloquia/Spring2016|Spring 2016]]<br />
<br />
[[Colloquia/Fall2015|Fall 2015]]<br />
<br />
[[Colloquia/Spring2014|Spring 2015]]<br />
<br />
[[Colloquia/Fall2014|Fall 2014]]<br />
<br />
[[Colloquia/Spring2014|Spring 2014]]<br />
<br />
[[Colloquia/Fall2013|Fall 2013]]<br />
<br />
[[Colloquia 2012-2013|Spring 2013]]<br />
<br />
[[Colloquia 2012-2013#Fall 2012|Fall 2012]]</div>Andreichttps://www.math.wisc.edu/wiki/index.php?title=Colloquia&diff=17521Colloquia2019-07-16T14:23:21Z<p>Andreic: </p>
<hr />
<div>= Mathematics Colloquium =<br />
<br />
All colloquia are on Fridays at 4:00 pm in Van Vleck B239, '''unless otherwise indicated'''.<br />
<br />
<br />
<br />
==Fall 2019==<br />
{| cellpadding="8"<br />
!align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|Sept 6<br />
|<br />
|<br />
|-<br />
|Sept 13<br />
| Yan Soibelman (Kansas State)<br />
|[[# TBA| TBA ]]<br />
| Caldararu<br />
|<br />
|-<br />
|Sept 16 '''Monday Room 911'''<br />
| Alicia Dickenstein (Buenos Aires)<br />
|[[# TBA| TBA ]]<br />
| Craciun<br />
|<br />
|-<br />
|Sept 20<br />
| Jianfeng Lu (Duke)<br />
|[[#TBA | TBA]]<br />
| Qin<br />
|<br />
|-<br />
|Sept 27<br />
|Elchnanan Mossel (MIT) Distinguished Lecture<br />
|-<br />
|Oct 4<br />
| Matt Baker (Georgia Tech)<br />
|<br />
|-<br />
|Oct 11<br />
|<br />
|-<br />
|Oct 18<br />
|<br />
|-<br />
|Oct 25<br />
|<br />
|-<br />
|Nov 1<br />
|Possibly reserved for job talk?<br />
|<br />
|-<br />
|Nov 8<br />
|Reserved for job talk<br />
|<br />
|-<br />
|Nov 15<br />
|Reserved for job talk<br />
|<br />
|-<br />
|Nov 22<br />
|Reserved for job talk<br />
|<br />
|-<br />
|Nov 29<br />
|Thanksgiving<br />
|<br />
|-<br />
|Dec 6<br />
|Reserved for job talk<br />
|<br />
|-<br />
|Dec 13<br />
|Reserved for job talk<br />
|<br />
|}<br />
<br />
==Spring 2020==<br />
<br />
{| cellpadding="8"<br />
!align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|<br />
|-<br />
|Jan 24<br />
|<br />
|-<br />
|Jan 31<br />
|<br />
|-<br />
|Feb 7<br />
|<br />
|-<br />
|Feb 14<br />
|<br />
|-<br />
|Feb 21<br />
|<br />
|-<br />
|Feb 28<br />
|<br />
|-<br />
|March 6<br />
|<br />
|-<br />
|March 13<br />
|<br />
|-<br />
|March 20<br />
|Spring break<br />
|<br />
|-<br />
|March 27<br />
|<br />
|-<br />
|April 3<br />
|<br />
|-<br />
|April 10<br />
| Sarah Koch (Michigan)<br />
|<br />
| Bruce (WIMAW)<br />
|-<br />
|April 17<br />
|<br />
|-<br />
|April 24<br />
|<br />
|-<br />
|May 1<br />
|Robert Lazarsfeld (Stony Brook)<br />
|Distinguished lecture<br />
|Erman<br />
|}<br />
<br />
== Abstracts ==<br />
<br />
===Person (Institution)===<br />
<br />
Title:<br />
<br />
Abstract:<br />
<br />
== Past Colloquia ==<br />
<br />
[[Colloquia/Blank|Blank]]<br />
<br />
[[Colloquia/Spring2019|Spring 2019]]<br />
<br />
[[Colloquia/Fall2018|Fall 2018]]<br />
<br />
[[Colloquia/Spring2018|Spring 2018]]<br />
<br />
[[Colloquia/Fall2017|Fall 2017]]<br />
<br />
[[Colloquia/Spring2017|Spring 2017]]<br />
<br />
[[Archived Fall 2016 Colloquia|Fall 2016]]<br />
<br />
[[Colloquia/Spring2016|Spring 2016]]<br />
<br />
[[Colloquia/Fall2015|Fall 2015]]<br />
<br />
[[Colloquia/Spring2014|Spring 2015]]<br />
<br />
[[Colloquia/Fall2014|Fall 2014]]<br />
<br />
[[Colloquia/Spring2014|Spring 2014]]<br />
<br />
[[Colloquia/Fall2013|Fall 2013]]<br />
<br />
[[Colloquia 2012-2013|Spring 2013]]<br />
<br />
[[Colloquia 2012-2013#Fall 2012|Fall 2012]]</div>Andreichttps://www.math.wisc.edu/wiki/index.php?title=Algebra_and_Algebraic_Geometry_Seminar_Spring_2019&diff=16835Algebra and Algebraic Geometry Seminar Spring 20192019-02-05T20:34:08Z<p>Andreic: /* Abstracts */</p>
<hr />
<div>The seminar meets on Fridays at 2:25 pm in room B235.<br />
<br />
Here is the schedule for [[Algebra and Algebraic Geometry Seminar Fall 2018 | the previous semester]], for [[Algebra and Algebraic Geometry Seminar Fall 2019 | the next semester]], and for [[Algebra and Algebraic Geometry Seminar | this semester]].<br />
<br />
==Algebra and Algebraic Geometry Mailing List==<br />
*Please join the [https://admin.lists.wisc.edu/index.php?p=11&l=ags AGS Mailing List] to hear about upcoming seminars, lunches, and other algebraic geometry events in the department (it is possible you must be on a math department computer to use this link).<br />
<br />
<br />
== Spring 2019 Schedule ==<br />
<br />
{| cellpadding="8"<br />
!align="left" | date<br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s) <br />
|-<br />
|January 25<br />
|[http://www.math.utah.edu/~smolkin/ Daniel Smolkin (Utah)]<br />
|Symbolic Powers in Rings of Positive Characteristic<br />
|Daniel<br />
|-<br />
|February 1<br />
|Juliette Bruce<br />
|Asymptotic Syzgies for Products of Projective Spaces<br />
|Local<br />
|-<br />
|February 8<br />
|[http://www.mit.edu/~ivogt/ Isabel Vogt (MIT)]<br />
| Low degree points on curves<br />
|Wanlin and Juliette<br />
|-<br />
|February 15<br />
|Pavlo Pylyavskyy (U. Minn)<br />
|Zamolodchikov periodicity and integrability<br />
|Paul Terwilliger<br />
|-<br />
|February 22<br />
|Michael Brown (Wisconsin)<br />
|Chern-Weil theory for matrix factorizations<br />
|Local<br />
|-<br />
|March 1<br />
|Shamgar Gurevich (Wisconsin)<br />
|Harmonic Analysis on GLn over finite fields, and Random Walks<br />
|Local<br />
|-<br />
|March 8<br />
|Jay Kopper (UIC)<br />
|TBD<br />
|Daniel<br />
|-<br />
|March 15<br />
|TBD<br />
|TBD<br />
|TBD<br />
|-<br />
|March 22<br />
|No Meeting<br />
|Spring Break<br />
|TBD<br />
|-<br />
|March 29<br />
|[https://math.berkeley.edu/~ceur/ Chris Eur (UC Berkeley)]<br />
|TBD<br />
|Daniel<br />
|-<br />
|April 5<br />
|TBD<br />
|TBD<br />
|TBD<br />
|-<br />
|April 12<br />
|[http://www-personal.umich.edu/~ecanton/ Eric Canton (Michigan)]<br />
|TBD<br />
|Michael<br />
|-<br />
|April 19<br />
|[http://www-personal.umich.edu/~grifo/ Elo&iacute;sa Grifo (Michigan)]<br />
|TBD<br />
|TBD<br />
|-<br />
|April 26<br />
|TBD<br />
|TBD<br />
|TBD<br />
|-<br />
|May 3<br />
|TBD<br />
|TBD<br />
|TBD<br />
|}<br />
<br />
== Abstracts ==<br />
<br />
===Daniel Smolkin===<br />
'''Symbolic Powers in Rings of Positive Characteristic'''<br />
<br />
The n-th power of an ideal is easy to compute, though difficult to describe geometrically. In contrast, symbolic powers of ideals are difficult to compute while having a natural geometric description. In this talk, I will describe how to compare ordinary and symbolic powers of ideals using the techniques of positive-characteristic commutative algebra, especially in toric rings and Hibi rings. This is based on joint work with Javier Carvajal-Rojas, Janet Page, and Kevin Tucker. Graduate students are encouraged to attend!<br />
<br />
===Juliette Bruce===<br />
<br />
'''Title: Asymptotic Syzygies for Products of Projective Spaces'''<br />
<br />
I will discuss results describing the asymptotic syzygies of products of projective space, in the vein of the explicit methods of Ein, Erman, and Lazarsfeld’s non-vanishing results on projective space.<br />
<br />
===Isabel Vogt===<br />
<br />
'''Title: Low degree points on curves'''<br />
<br />
In this talk we will discuss an arithmetic analogue of the gonality of a curve over a number field: the smallest positive integer e such that the points of residue degree bounded by e are infinite. By work of Faltings, Harris--Silverman and Abramovich--Harris, it is well-understood when this invariant is 1, 2, or 3; by work of Debarre--Fahlaoui these criteria do not generalize to e at least 4. We will study this invariant using the auxiliary geometry of a surface containing the curve and devote particular attention to scenarios under which we can guarantee that this invariant is actually equal to the gonality . This is joint work with Geoffrey Smith.<br />
<br />
===Pavlo Pylyavskyy===<br />
<br />
'''Zamolodchikov periodicity and integrability'''<br />
<br />
T-systems are certain discrete dynamical systems associated with quivers. They appear in several different contexts: quantum affine algebras and Yangians, commuting transfer matrices of vertex models, character theory of quantum groups, analytic Bethe ansatz, Wronskian-Casoratian duality in ODE, gauge/string theories, etc. Periodicity of certain T-systems was the main conjecture in the area until it was proven by Keller in 2013 using cluster categories. In this work we completely classify periodic T-systems, which turn out to consist of 5 infinite families and 4 exceptional cases, only one of the infinite families being known previously. We then proceed to classify T-systems that exhibit two forms of integrability: linearization and zero algebraic entropy. All three classifications rely on reduction of the problem to study of commuting Cartan matrices, either of finite or affine types. The finite type classification was obtained by Stembridge in his study of Kazhdan-Lusztig theory for dihedral groups, the other two classifications are new. This is joint work with Pavel Galashin. <br />
<br />
===Shamgar Gurevich===<br />
<br />
'''Harmonic Analysis on GLn over finite fields, and Random Walks'''<br />
<br />
There are many formulas that express interesting properties of a group G in terms of sums over its characters. For evaluating or estimating these sums, one of the most salient quantities to understand is the {\it character ratio}: <br />
<br />
$$<br />
trace(\rho(g))/dim(\rho),<br />
$$<br />
<br />
for an irreducible representation $\rho$ of G and an element g of G. For example, Diaconis and Shahshahani stated a formula of this type for analyzing G-biinvariant random walks on G. It turns out that, for classical groups G over finite fields (which provide most examples of finite simple groups), there is a natural invariant of representations that provides strong information on the character ratio. We call this invariant {\it rank}. This talk will discuss the notion of rank for GLn over finite fields, and apply the results to random walks. This is joint work with Roger Howe (Yale and Texas AM).</div>Andreichttps://www.math.wisc.edu/wiki/index.php?title=Algebra_and_Algebraic_Geometry_Seminar_Spring_2019&diff=16834Algebra and Algebraic Geometry Seminar Spring 20192019-02-05T20:33:17Z<p>Andreic: /* Spring 2019 Schedule */</p>
<hr />
<div>The seminar meets on Fridays at 2:25 pm in room B235.<br />
<br />
Here is the schedule for [[Algebra and Algebraic Geometry Seminar Fall 2018 | the previous semester]], for [[Algebra and Algebraic Geometry Seminar Fall 2019 | the next semester]], and for [[Algebra and Algebraic Geometry Seminar | this semester]].<br />
<br />
==Algebra and Algebraic Geometry Mailing List==<br />
*Please join the [https://admin.lists.wisc.edu/index.php?p=11&l=ags AGS Mailing List] to hear about upcoming seminars, lunches, and other algebraic geometry events in the department (it is possible you must be on a math department computer to use this link).<br />
<br />
<br />
== Spring 2019 Schedule ==<br />
<br />
{| cellpadding="8"<br />
!align="left" | date<br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s) <br />
|-<br />
|January 25<br />
|[http://www.math.utah.edu/~smolkin/ Daniel Smolkin (Utah)]<br />
|Symbolic Powers in Rings of Positive Characteristic<br />
|Daniel<br />
|-<br />
|February 1<br />
|Juliette Bruce<br />
|Asymptotic Syzgies for Products of Projective Spaces<br />
|Local<br />
|-<br />
|February 8<br />
|[http://www.mit.edu/~ivogt/ Isabel Vogt (MIT)]<br />
| Low degree points on curves<br />
|Wanlin and Juliette<br />
|-<br />
|February 15<br />
|Pavlo Pylyavskyy (U. Minn)<br />
|Zamolodchikov periodicity and integrability<br />
|Paul Terwilliger<br />
|-<br />
|February 22<br />
|Michael Brown (Wisconsin)<br />
|Chern-Weil theory for matrix factorizations<br />
|Local<br />
|-<br />
|March 1<br />
|Shamgar Gurevich (Wisconsin)<br />
|Harmonic Analysis on GLn over finite fields, and Random Walks<br />
|Local<br />
|-<br />
|March 8<br />
|Jay Kopper (UIC)<br />
|TBD<br />
|Daniel<br />
|-<br />
|March 15<br />
|TBD<br />
|TBD<br />
|TBD<br />
|-<br />
|March 22<br />
|No Meeting<br />
|Spring Break<br />
|TBD<br />
|-<br />
|March 29<br />
|[https://math.berkeley.edu/~ceur/ Chris Eur (UC Berkeley)]<br />
|TBD<br />
|Daniel<br />
|-<br />
|April 5<br />
|TBD<br />
|TBD<br />
|TBD<br />
|-<br />
|April 12<br />
|[http://www-personal.umich.edu/~ecanton/ Eric Canton (Michigan)]<br />
|TBD<br />
|Michael<br />
|-<br />
|April 19<br />
|[http://www-personal.umich.edu/~grifo/ Elo&iacute;sa Grifo (Michigan)]<br />
|TBD<br />
|TBD<br />
|-<br />
|April 26<br />
|TBD<br />
|TBD<br />
|TBD<br />
|-<br />
|May 3<br />
|TBD<br />
|TBD<br />
|TBD<br />
|}<br />
<br />
== Abstracts ==<br />
<br />
===Daniel Smolkin===<br />
'''Symbolic Powers in Rings of Positive Characteristic'''<br />
<br />
The n-th power of an ideal is easy to compute, though difficult to describe geometrically. In contrast, symbolic powers of ideals are difficult to compute while having a natural geometric description. In this talk, I will describe how to compare ordinary and symbolic powers of ideals using the techniques of positive-characteristic commutative algebra, especially in toric rings and Hibi rings. This is based on joint work with Javier Carvajal-Rojas, Janet Page, and Kevin Tucker. Graduate students are encouraged to attend!<br />
<br />
===Juliette Bruce===<br />
<br />
'''Title: Asymptotic Syzygies for Products of Projective Spaces'''<br />
<br />
I will discuss results describing the asymptotic syzygies of products of projective space, in the vein of the explicit methods of Ein, Erman, and Lazarsfeld’s non-vanishing results on projective space.<br />
<br />
===Isabel Vogt===<br />
<br />
'''Title: Low degree points on curves'''<br />
<br />
In this talk we will discuss an arithmetic analogue of the gonality of a curve over a number field: the smallest positive integer e such that the points of residue degree bounded by e are infinite. By work of Faltings, Harris--Silverman and Abramovich--Harris, it is well-understood when this invariant is 1, 2, or 3; by work of Debarre--Fahlaoui these criteria do not generalize to e at least 4. We will study this invariant using the auxiliary geometry of a surface containing the curve and devote particular attention to scenarios under which we can guarantee that this invariant is actually equal to the gonality . This is joint work with Geoffrey Smith.<br />
<br />
<br />
===Shamgar Gurevich===<br />
<br />
'''Harmonic Analysis on GLn over finite fields, and Random Walks'''<br />
<br />
There are many formulas that express interesting properties of a group G in terms of sums over its characters. For evaluating or estimating these sums, one of the most salient quantities to understand is the {\it character ratio}: <br />
<br />
$$<br />
trace(\rho(g))/dim(\rho),<br />
$$<br />
<br />
for an irreducible representation $\rho$ of G and an element g of G. For example, Diaconis and Shahshahani stated a formula of this type for analyzing G-biinvariant random walks on G. It turns out that, for classical groups G over finite fields (which provide most examples of finite simple groups), there is a natural invariant of representations that provides strong information on the character ratio. We call this invariant {\it rank}. This talk will discuss the notion of rank for GLn over finite fields, and apply the results to random walks. This is joint work with Roger Howe (Yale and Texas AM).</div>Andreichttps://www.math.wisc.edu/wiki/index.php?title=Colloquia&diff=16813Colloquia2019-02-04T18:24:12Z<p>Andreic: </p>
<hr />
<div>= Mathematics Colloquium =<br />
<br />
All colloquia are on Fridays at 4:00 pm in Van Vleck B239, '''unless otherwise indicated'''.<br />
<br />
==Spring 2019==<br />
<br />
{| cellpadding="8"<br />
!align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|Jan 25 '''Room 911'''<br />
| [http://www.users.miamioh.edu/randrib/ Beata Randrianantoanina] (Miami University Ohio) WIMAW<br />
|[[#Beata Randrianantoanina (Miami University Ohio) | Some nonlinear problems in the geometry of Banach spaces and their applications ]]<br />
| Tullia Dymarz<br />
|<br />
|-<br />
|Jan 30 '''Wednesday'''<br />
| Talk rescheduled to Feb 15<br />
|<br />
|-<br />
|Jan 31 '''Thursday'''<br />
| Talk rescheduled to Feb 13<br />
|<br />
|-<br />
|Feb 1<br />
| Talk cancelled due to weather<br />
|<br />
| <br />
|<br />
|-<br />
|Feb 5 '''Tuesday'''<br />
| [http://www.math.tamu.edu/~alexei.poltoratski/ Alexei Poltoratski] (Texas A&M University)<br />
|[[#Alexei Poltoratski (Texas A&M)| Completeness of exponentials: Beurling-Malliavin and type problems ]]<br />
| Denisov<br />
|<br />
|-<br />
|Feb 8<br />
| [https://sites.math.northwestern.edu/~anaber/ Aaron Naber] (Northwestern)<br />
|[[#Aaron Naber (Northwestern) | A structure theory for spaces with lower Ricci curvature bounds ]]<br />
| Street<br />
|<br />
|-<br />
|Feb 11 '''Monday'''<br />
| [https://www2.bc.edu/david-treumann/materials.html David Treumann] (Boston College)<br />
|[[#David Treumann (Boston College) | Twisting things in topology and symplectic topology by pth powers ]]<br />
| Caldararu<br />
|<br />
|-<br />
| Feb 13 '''Wednesday'''<br />
| [http://www.math.tamu.edu/~dbaskin/ Dean Baskin] (Texas A&M)<br />
|[[#Dean Baskin (Texas A&M) | Radiation fields for wave equations ]]<br />
| Street<br />
<br />
|-<br />
| Feb 15 <br />
| [https://services.math.duke.edu/~pierce/ Lillian Pierce] (Duke University)<br />
| [[#Lillian Pierce (Duke University) | Short character sums ]]<br />
| Boston and Street<br />
|<br />
|-<br />
|Feb 22<br />
| [https://people.math.osu.edu/cueto.5/ Angelica Cueto] (Ohio State)<br />
|[[# TBA| TBA ]]<br />
| Erman and Corey<br />
|<br />
|-<br />
|March 4<br />
| [http://www-users.math.umn.edu/~sverak/ Vladimir Sverak] (Minnesota) Wasow lecture<br />
|[[# TBA| TBA ]]<br />
| Kim<br />
|<br />
|-<br />
|March 8<br />
| [https://orion.math.iastate.edu/jmccullo/index.html Jason McCullough] (Iowa State)<br />
|[[# TBA| TBA ]]<br />
| Erman<br />
|<br />
|-<br />
|March 15<br />
| Maksym Radziwill (Caltech)<br />
|[[# TBA| TBA ]]<br />
| Marshall<br />
|<br />
|-<br />
|March 29<br />
| Jennifer Park (OSU)<br />
|[[# TBA| TBA ]]<br />
| Marshall<br />
|<br />
|-<br />
|April 5<br />
| Ju-Lee Kim (MIT)<br />
|[[# TBA| TBA ]]<br />
| Gurevich<br />
|<br />
|-<br />
|April 12<br />
| Evitar Procaccia (TAMU)<br />
|[[# TBA| TBA ]]<br />
| Gurevich<br />
|<br />
|-<br />
|April 19<br />
| [http://www.math.rice.edu/~jkn3/ Jo Nelson] (Rice University)<br />
|[[# TBA| TBA ]]<br />
| Jean-Luc<br />
|<br />
|-<br />
|April 26<br />
| [https://www.brown.edu/academics/applied-mathematics/faculty/kavita-ramanan/home Kavita Ramanan] (Brown University)<br />
|[[# TBA| TBA ]]<br />
| WIMAW<br />
|<br />
|-<br />
|May 3<br />
| Tomasz Przebinda (Oklahoma)<br />
|[[# TBA| TBA ]]<br />
| Gurevich<br />
|<br />
|}<br />
<br />
== Abstracts ==<br />
<br />
===Beata Randrianantoanina (Miami University Ohio)===<br />
<br />
Title: Some nonlinear problems in the geometry of Banach spaces and their applications.<br />
<br />
Abstract: Nonlinear problems in the geometry of Banach spaces have been studied since the inception of the field. In this talk I will outline some of the history, some of modern applications, and some open directions of research. The talk will be accessible to graduate students of any field of mathematics.<br />
<br />
===Lillian Pierce (Duke University)===<br />
<br />
Title: Short character sums <br />
<br />
Abstract: A surprisingly diverse array of problems in analytic number theory have at their heart a problem of bounding (from above) an exponential sum, or its multiplicative cousin, a so-called character sum. For example, both understanding the Riemann zeta function or Dirichlet L-functions inside the critical strip, and also counting solutions to Diophantine equations via the circle method or power sieve methods, involve bounding such sums. In general, the sums of interest fall into one of two main regimes: complete sums or incomplete sums, with this latter regime including in particular “short sums.” Short sums are particularly useful, and particularly resistant to almost all known methods. In this talk, we will see what makes a sum “short,” sketch why it would be incredibly powerful to understand short sums, and discuss a curious proof from the 1950’s which is still the best way we know to bound short sums. We will end by describing new work which extends the ideas of this curious proof to bound short sums in much more general situations.<br />
<br />
===David Treumann (Boston College)===<br />
<br />
Title: Twisting things in topology and symplectic topology by pth powers<br />
<br />
Abstract: There's an old and popular analogy between circles and finite fields. I'll describe some constructions you can make in Lagrangian Floer theory and in microlocal sheaf theory by taking this analogy extremely literally, the main ingredient is an "F-field." An F-field on a manifold M is a local system of algebraically closed fields of characteristic p. When M is symplectic, maybe an F-field should remind you of a B-field, it can be used to change the Fukaya category in about the same way. On M = S^1 times R^3, this version of the Fukaya category is related to Deligne-Lusztig theory, and I found something like a cluster structure on the Deligne-Lusztig pairing varieties by studying it. On M = S^1 times S^1, Yanki Lekili and I have found that this version of the Fukaya category is related to the equal-characteristic version of the Fargues-Fontaine curve; the relationship is homological mirror symmetry.<br />
<br />
===Dean Baskin (Texas A&M)===<br />
<br />
Title: Radiation fields for wave equations<br />
<br />
Abstract: Radiation fields are rescaled limits of solutions of wave equations near "null infinity" and capture the radiation pattern seen by a distant observer. They are intimately connected with the Fourier and Radon transforms and with scattering theory. In this talk, I will define and discuss radiation fields in a few contexts, with an emphasis on spacetimes that look flat near infinity. The main result is a connection between the asymptotic behavior of the radiation field and a family of quantum objects on an associated asymptotically hyperbolic space.<br />
<br />
===Jianfeng Lu (Duke University)===<br />
<br />
Title: Density fitting: Analysis, algorithm and applications<br />
<br />
Abstract: Density fitting considers the low-rank approximation of pair products of eigenfunctions of Hamiltonian operators. It is a very useful tool with many applications in electronic structure theory. In this talk, we will discuss estimates of upper bound of the numerical rank of the pair products of eigenfunctions. We will also introduce the interpolative separable density fitting (ISDF) algorithm, which reduces the computational scaling of the low-rank approximation and can be used for efficient algorithms for electronic structure calculations. Based on joint works with Chris Sogge, Stefan Steinerberger, Kyle Thicke, and Lexing Ying.<br />
<br />
===Alexei Poltoratski (Texas A&M)===<br />
<br />
Title: Completeness of exponentials: Beurling-Malliavin and type problems<br />
<br />
Abstract: This talk is devoted to two old problems of harmonic analysis mentioned in the title. Both<br />
problems ask when a family of complex exponentials is complete (spans) an L^2-space. The Beruling-Malliavin<br />
problem was solved in the early 1960s and I will present its classical solution along with modern generalizations<br />
and applications. I will then discuss history and recent progress in the type problem, which stood open for<br />
more than 70 years.<br />
<br />
===Aaron Naber (Northwestern)===<br />
<br />
Title: A structure theory for spaces with lower Ricci curvature bounds.<br />
<br />
Abstract: One should view manifolds (M^n,g) with lower Ricci curvature bounds as being those manifolds with a well behaved analysis, a point which can be rigorously stated. It thus becomes a natural question, how well behaved or badly behaved can such spaces be? This is a nonlinear analogue to asking how degenerate can a subharmonic or plurisubharmonic function look like. In this talk we give an essentially sharp answer to this question. The talk will require little background, and our time will be spent on understanding the basic statements and examples. The work discussed is joint with Cheeger, Jiang and with Li.<br />
<br />
== Past Colloquia ==<br />
<br />
[[Colloquia/Blank|Blank]]<br />
<br />
[[Colloquia/Fall2018|Fall 2018]]<br />
<br />
[[Colloquia/Spring2018|Spring 2018]]<br />
<br />
[[Colloquia/Fall2017|Fall 2017]]<br />
<br />
[[Colloquia/Spring2017|Spring 2017]]<br />
<br />
[[Archived Fall 2016 Colloquia|Fall 2016]]<br />
<br />
[[Colloquia/Spring2016|Spring 2016]]<br />
<br />
[[Colloquia/Fall2015|Fall 2015]]<br />
<br />
[[Colloquia/Spring2014|Spring 2015]]<br />
<br />
[[Colloquia/Fall2014|Fall 2014]]<br />
<br />
[[Colloquia/Spring2014|Spring 2014]]<br />
<br />
[[Colloquia/Fall2013|Fall 2013]]<br />
<br />
[[Colloquia 2012-2013|Spring 2013]]<br />
<br />
[[Colloquia 2012-2013#Fall 2012|Fall 2012]]</div>Andreichttps://www.math.wisc.edu/wiki/index.php?title=Colloquia&diff=16810Colloquia2019-02-04T14:57:31Z<p>Andreic: /* Spring 2019 */</p>
<hr />
<div>= Mathematics Colloquium =<br />
<br />
All colloquia are on Fridays at 4:00 pm in Van Vleck B239, '''unless otherwise indicated'''.<br />
<br />
==Spring 2019==<br />
<br />
{| cellpadding="8"<br />
!align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|Jan 25 '''Room 911'''<br />
| [http://www.users.miamioh.edu/randrib/ Beata Randrianantoanina] (Miami University Ohio) WIMAW<br />
|[[#Beata Randrianantoanina (Miami University Ohio) | Some nonlinear problems in the geometry of Banach spaces and their applications ]]<br />
| Tullia Dymarz<br />
|<br />
|-<br />
|Jan 30 '''Wednesday'''<br />
| Talk rescheduled to Feb 15<br />
|<br />
|-<br />
|Jan 31 '''Thursday'''<br />
| Talk rescheduled to Feb 13<br />
|<br />
|-<br />
|Feb 1<br />
| Talk cancelled due to weather<br />
|<br />
| <br />
|<br />
|-<br />
|Feb 5 '''Tuesday'''<br />
| [http://www.math.tamu.edu/~alexei.poltoratski/ Alexei Poltoratski] (Texas A&M University)<br />
|[[#Alexei Poltoratski (Texas A&M)| Completeness of exponentials: Beurling-Malliavin and type problems ]]<br />
| Denisov<br />
|<br />
|-<br />
|Feb 8<br />
| [https://sites.math.northwestern.edu/~anaber/ Aaron Naber] (Northwestern)<br />
|[[#Aaron Naber (Northwestern) | A structure theory for spaces with lower Ricci curvature bounds ]]<br />
| Street<br />
|<br />
|-<br />
|Feb 11 '''Monday'''<br />
| [https://www2.bc.edu/david-treumann/materials.html David Treumann] (Boston College)<br />
|[[#David Treumann (Boston College) | TBA ]]<br />
| Caldararu<br />
|<br />
|-<br />
| Feb 13 '''Wednesday'''<br />
| [http://www.math.tamu.edu/~dbaskin/ Dean Baskin] (Texas A&M)<br />
|[[#Dean Baskin (Texas A&M) | Radiation fields for wave equations ]]<br />
| Street<br />
<br />
|-<br />
| Feb 15 <br />
| [https://services.math.duke.edu/~pierce/ Lillian Pierce] (Duke University)<br />
| [[#Lillian Pierce (Duke University) | Short character sums ]]<br />
| Boston and Street<br />
|<br />
|-<br />
|Feb 22<br />
| [https://people.math.osu.edu/cueto.5/ Angelica Cueto] (Ohio State)<br />
|[[# TBA| TBA ]]<br />
| Erman and Corey<br />
|<br />
|-<br />
|March 4<br />
| [http://www-users.math.umn.edu/~sverak/ Vladimir Sverak] (Minnesota) Wasow lecture<br />
|[[# TBA| TBA ]]<br />
| Kim<br />
|<br />
|-<br />
|March 8<br />
| [https://orion.math.iastate.edu/jmccullo/index.html Jason McCullough] (Iowa State)<br />
|[[# TBA| TBA ]]<br />
| Erman<br />
|<br />
|-<br />
|March 15<br />
| Maksym Radziwill (Caltech)<br />
|[[# TBA| TBA ]]<br />
| Marshall<br />
|<br />
|-<br />
|March 29<br />
| Jennifer Park (OSU)<br />
|[[# TBA| TBA ]]<br />
| Marshall<br />
|<br />
|-<br />
|April 5<br />
| Ju-Lee Kim (MIT)<br />
|[[# TBA| TBA ]]<br />
| Gurevich<br />
|<br />
|-<br />
|April 12<br />
| Evitar Procaccia (TAMU)<br />
|[[# TBA| TBA ]]<br />
| Gurevich<br />
|<br />
|-<br />
|April 19<br />
| [http://www.math.rice.edu/~jkn3/ Jo Nelson] (Rice University)<br />
|[[# TBA| TBA ]]<br />
| Jean-Luc<br />
|<br />
|-<br />
|April 26<br />
| [https://www.brown.edu/academics/applied-mathematics/faculty/kavita-ramanan/home Kavita Ramanan] (Brown University)<br />
|[[# TBA| TBA ]]<br />
| WIMAW<br />
|<br />
|-<br />
|May 3<br />
| Tomasz Przebinda (Oklahoma)<br />
|[[# TBA| TBA ]]<br />
| Gurevich<br />
|<br />
|}<br />
<br />
== Abstracts ==<br />
<br />
===Beata Randrianantoanina (Miami University Ohio)===<br />
<br />
Title: Some nonlinear problems in the geometry of Banach spaces and their applications.<br />
<br />
Abstract: Nonlinear problems in the geometry of Banach spaces have been studied since the inception of the field. In this talk I will outline some of the history, some of modern applications, and some open directions of research. The talk will be accessible to graduate students of any field of mathematics.<br />
<br />
===Lillian Pierce (Duke University)===<br />
<br />
Title: Short character sums <br />
<br />
Abstract: A surprisingly diverse array of problems in analytic number theory have at their heart a problem of bounding (from above) an exponential sum, or its multiplicative cousin, a so-called character sum. For example, both understanding the Riemann zeta function or Dirichlet L-functions inside the critical strip, and also counting solutions to Diophantine equations via the circle method or power sieve methods, involve bounding such sums. In general, the sums of interest fall into one of two main regimes: complete sums or incomplete sums, with this latter regime including in particular “short sums.” Short sums are particularly useful, and particularly resistant to almost all known methods. In this talk, we will see what makes a sum “short,” sketch why it would be incredibly powerful to understand short sums, and discuss a curious proof from the 1950’s which is still the best way we know to bound short sums. We will end by describing new work which extends the ideas of this curious proof to bound short sums in much more general situations.<br />
<br />
===Dean Baskin (Texas A&M)===<br />
<br />
Title: Radiation fields for wave equations<br />
<br />
Abstract: Radiation fields are rescaled limits of solutions of wave equations near "null infinity" and capture the radiation pattern seen by a distant observer. They are intimately connected with the Fourier and Radon transforms and with scattering theory. In this talk, I will define and discuss radiation fields in a few contexts, with an emphasis on spacetimes that look flat near infinity. The main result is a connection between the asymptotic behavior of the radiation field and a family of quantum objects on an associated asymptotically hyperbolic space.<br />
<br />
===Jianfeng Lu (Duke University)===<br />
<br />
Title: Density fitting: Analysis, algorithm and applications<br />
<br />
Abstract: Density fitting considers the low-rank approximation of pair products of eigenfunctions of Hamiltonian operators. It is a very useful tool with many applications in electronic structure theory. In this talk, we will discuss estimates of upper bound of the numerical rank of the pair products of eigenfunctions. We will also introduce the interpolative separable density fitting (ISDF) algorithm, which reduces the computational scaling of the low-rank approximation and can be used for efficient algorithms for electronic structure calculations. Based on joint works with Chris Sogge, Stefan Steinerberger, Kyle Thicke, and Lexing Ying.<br />
<br />
===Alexei Poltoratski (Texas A&M)===<br />
<br />
Title: Completeness of exponentials: Beurling-Malliavin and type problems<br />
<br />
Abstract: This talk is devoted to two old problems of harmonic analysis mentioned in the title. Both<br />
problems ask when a family of complex exponentials is complete (spans) an L^2-space. The Beruling-Malliavin<br />
problem was solved in the early 1960s and I will present its classical solution along with modern generalizations<br />
and applications. I will then discuss history and recent progress in the type problem, which stood open for<br />
more than 70 years.<br />
<br />
===Aaron Naber (Northwestern)===<br />
<br />
Title: A structure theory for spaces with lower Ricci curvature bounds.<br />
<br />
Abstract: One should view manifolds (M^n,g) with lower Ricci curvature bounds as being those manifolds with a well behaved analysis, a point which can be rigorously stated. It thus becomes a natural question, how well behaved or badly behaved can such spaces be? This is a nonlinear analogue to asking how degenerate can a subharmonic or plurisubharmonic function look like. In this talk we give an essentially sharp answer to this question. The talk will require little background, and our time will be spent on understanding the basic statements and examples. The work discussed is joint with Cheeger, Jiang and with Li.<br />
<br />
== Past Colloquia ==<br />
<br />
[[Colloquia/Blank|Blank]]<br />
<br />
[[Colloquia/Fall2018|Fall 2018]]<br />
<br />
[[Colloquia/Spring2018|Spring 2018]]<br />
<br />
[[Colloquia/Fall2017|Fall 2017]]<br />
<br />
[[Colloquia/Spring2017|Spring 2017]]<br />
<br />
[[Archived Fall 2016 Colloquia|Fall 2016]]<br />
<br />
[[Colloquia/Spring2016|Spring 2016]]<br />
<br />
[[Colloquia/Fall2015|Fall 2015]]<br />
<br />
[[Colloquia/Spring2014|Spring 2015]]<br />
<br />
[[Colloquia/Fall2014|Fall 2014]]<br />
<br />
[[Colloquia/Spring2014|Spring 2014]]<br />
<br />
[[Colloquia/Fall2013|Fall 2013]]<br />
<br />
[[Colloquia 2012-2013|Spring 2013]]<br />
<br />
[[Colloquia 2012-2013#Fall 2012|Fall 2012]]</div>Andreichttps://www.math.wisc.edu/wiki/index.php?title=Algebra_and_Algebraic_Geometry_Seminar_Fall_2018&diff=16217Algebra and Algebraic Geometry Seminar Fall 20182018-10-16T03:30:23Z<p>Andreic: </p>
<hr />
<div>The seminar meets on Fridays at 2:25 pm in room B235.<br />
<br />
Here is the schedule for [[Algebra and Algebraic Geometry Seminar Spring 2018 | the previous semester]], [[Algebra and Algebraic Geometry Seminar Spring 2019 | the next semester]], and for [[Algebra and Algebraic Geometry Seminar | this semester]].<br />
<br />
==Algebra and Algebraic Geometry Mailing List==<br />
<br />
<br />
*Please join the [https://admin.lists.wisc.edu/index.php?p=11&l=ags AGS Mailing List] to hear about upcoming seminars, lunches, and other algebraic geometry events in the department (it is possible you must be on a math department computer to use this link).<br />
<br />
== Fall 2018 Schedule ==<br />
<br />
{| cellpadding="8"<br />
!align="left" | date<br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s) <br />
<br />
|-<br />
|September 7<br />
|Daniel Erman<br />
|Big Polynomial Rings<br />
|Local<br />
|-<br />
|September 14<br />
|Akhil Mathew (U Chicago)<br />
|Kaledin's noncommutative degeneration theorem and topological Hochschild homology<br />
|Andrei<br />
|-<br />
|September 21<br />
|Andrei Caldararu<br />
|Categorical Gromov-Witten invariants beyond genus 1<br />
|Local<br />
|-<br />
|September 28<br />
|Mark Walker (Nebraska)<br />
|Conjecture D for matrix factorizations<br />
|Michael and Daniel<br />
|-<br />
|October 5<br />
|-<br />
|-<br />
|-<br />
|-<br />
|October 12<br />
|Jose Rodriguez (Wisconsin)<br />
|TBD<br />
|Local<br />
|-<br />
|October 19<br />
|Oleksandr Tsymbaliuk (Yale)<br />
|Modified quantum difference Toda systems<br />
|Paul Terwilliger<br />
|-<br />
|October 26<br />
|[https://juliettebruce.github.io Juliette Bruce]<br />
|Covering Abelian Varieties and Effective Bertini<br />
|Local<br />
|-<br />
|November 2<br />
|[http://sites.nd.edu/b-taji/ Behrouz Taji] (Notre Dame)<br />
|TBD<br />
|Botong Wang<br />
|-<br />
|November 9<br />
|[http://www-personal.umich.edu/~rohitna/ Rohit Nagpal (Michigan)]<br />
|TBD<br />
|John WG<br />
|-<br />
|November 16<br />
|Wanlin Li<br />
|TBD<br />
|Local<br />
|-<br />
|November 23<br />
|Thanksgiving<br />
|No Seminar<br />
|<br />
|-<br />
|November 30<br />
|John Wiltshire-Gordon<br />
|TBD<br />
|Local<br />
|-<br />
|December 7<br />
|Michael Brown<br />
|TBD<br />
|Local<br />
|-<br />
|December 14<br />
|TBD (this date is now open again!)<br />
|TBD<br />
|<br />
|-<br />
|}<br />
<br />
== Abstracts ==<br />
<br />
===Akhil Mathew===<br />
<br />
'''Title: Kaledin's noncommutative degeneration theorem and topological Hochschild homology'''<br />
<br />
For a smooth proper variety over a field of characteristic<br />
zero, the Hodge-to-de Rham spectral sequence (relating the cohomology<br />
of differential forms to de Rham cohomology) is well-known to<br />
degenerate, via Hodge theory. A "noncommutative" version of this<br />
theorem has been proved by Kaledin for smooth proper dg categories<br />
over a field of characteristic zero, based on the technique of<br />
reduction mod p. I will describe a short proof of this theorem using<br />
the theory of topological Hochschild homology, which provides a<br />
canonical one-parameter deformation of Hochschild homology in<br />
characteristic p.<br />
<br />
===Andrei Caldararu===<br />
'''Categorical Gromov-Witten invariants beyond genus 1'''<br />
<br />
In a seminal work from 2005 Kevin Costello defined numerical invariants associated to a <br />
Calabi-Yau A-infinity category. These invariants are supposed to generalize the classical<br />
Gromov-Witten invariants (counting curves in a target symplectic manifold) when the category<br />
is taken to be the Fukaya category. In my talk I shall describe some of the ideas involved in Costello's<br />
approach and recent progress (with Junwu Tu) on extending computations of these invariants<br />
past genus 1.<br />
<br />
===Mark Walker===<br />
'''Conjecture D for matrix factorizations'''<br />
<br />
Matrix factorizations form a dg category whose associated homotopy category is equivalent to the stable category of maximum Cohen-Macaulay modules over a hypersurface ring. In the isolated singularity case, the dg category of matrix factorizations is "smooth" and "proper" --- non-commutative analogues of the same-named properties of algebraic varieties. In general, for any smooth and proper dg category, there exist non-commutative analogues of Grothendieck's Standard Conjectures for cycles on smooth and projective varieties. In particular, the non-commutative version of Standard Conjecture D predicts that numerical equivalence and homological equivalence coincide for such a dg category. Recently, Michael Brown and I have proven the non-commutative analogue of Conjecture D for the category of matrix factorizations of an isolated singularity over a field of characteristic 0. In this talk, I will describe our theorem in more detail and give a sense of its proof.<br />
<br />
===Oleksandr Tsymbaliuk===<br />
'''Modified quantum difference Toda systems'''<br />
<br />
The q-version of a Toda system associated with any Lie algebra was introduced independently by Etingof and Sevostyanov in 1999. In this talk, we shall discuss the generalization of this construction which naturally produces a family of 3^{rk(g)-1} similar integrable systems. One of the key ingredients in the proof is played by the fermionic formula for the J-factors (defined as pairing of two Whittaker vectors in Verma modules), due to Feigin-Feigin-Jimbo-Miwa-Mukhin. In types A and C, our construction admits an alternative presentation via local Lax matrices, similar to the classical construction of Faddeev-Takhtajan for the classical type A Toda system. Finally, we shall discuss the geometric interpretation of Whittaker vectors in type A. <br />
<br />
This talk is based on the joint work with M. Finkelberg and R. Gonin.<br />
<br />
===Juliette Bruce===<br />
'''Covering Abelian Varieties and Effective Bertini'''<br />
<br />
I will discuss recent work showing that every abelian variety is covered by a Jacobian whose dimension is bounded. This is joint with Wanlin Li.</div>Andreichttps://www.math.wisc.edu/wiki/index.php?title=Algebra_and_Algebraic_Geometry_Seminar_Fall_2018&diff=16005Algebra and Algebraic Geometry Seminar Fall 20182018-09-17T21:38:34Z<p>Andreic: /* Abstracts */</p>
<hr />
<div>The seminar meets on Fridays at 2:25 pm in room B235.<br />
<br />
Here is the schedule for [[Algebra and Algebraic Geometry Seminar Spring 2018 | the previous semester]], [[Algebra and Algebraic Geometry Seminar Spring 2019 | the next semester]], and for [[Algebra and Algebraic Geometry Seminar | this semester]].<br />
<br />
==Algebra and Algebraic Geometry Mailing List==<br />
<br />
<br />
*Please join the [https://admin.lists.wisc.edu/index.php?p=11&l=ags AGS Mailing List] to hear about upcoming seminars, lunches, and other algebraic geometry events in the department (it is possible you must be on a math department computer to use this link).<br />
<br />
== Fall 2018 Schedule ==<br />
<br />
{| cellpadding="8"<br />
!align="left" | date<br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s) <br />
<br />
|-<br />
|September 7<br />
|Daniel Erman<br />
|Big Polynomial Rings<br />
|Local<br />
|-<br />
|September 14<br />
|Akhil Mathew (U Chicago)<br />
|Kaledin's noncommutative degeneration theorem and topological Hochschild homology<br />
|Andrei<br />
|-<br />
|September 21<br />
|Andrei Caldararu<br />
|Categorical Gromov-Witten invariants beyond genus 1<br />
|Local<br />
|-<br />
|September 28<br />
|Mark Walker (Nebraska)<br />
|TBD<br />
|Michael and Daniel<br />
|-<br />
|October 5<br />
|-<br />
|-<br />
|-<br />
|-<br />
|October 12<br />
|Jose Rodriguez (Wisconsin)<br />
|TBD<br />
|Local<br />
|-<br />
|October 19<br />
|Oleksandr Tsymbaliuk (Yale)<br />
|TBD<br />
|Paul Terwilliger<br />
|-<br />
|October 26<br />
|<br />
|<br />
|<br />
|-<br />
|November 2<br />
|Behrouz Taji (Notre Dame)<br />
|TBD<br />
|Botong Wang<br />
|-<br />
|November 9<br />
|Juliette Bruce<br />
|TBD<br />
|Local<br />
|-<br />
|November 16<br />
|Wanlin Li<br />
|TBD<br />
|Local<br />
|-<br />
|November 23<br />
|Thanksgiving<br />
|No Seminar<br />
|<br />
|-<br />
|November 30<br />
|[http://www-personal.umich.edu/~grifo/ Eloísa Grifo] (Michigan)<br />
|TBD<br />
|Daniel<br />
|-<br />
|December 7<br />
|Michael Brown<br />
|TBD<br />
|Local<br />
|-<br />
|December 14<br />
|John Wiltshire-Gordon<br />
|TBD<br />
|Local<br />
|-<br />
|}<br />
<br />
== Abstracts ==<br />
<br />
===Akhil Mathew===<br />
<br />
'''Title: Kaledin's noncommutative degeneration theorem and topological Hochschild homology'''<br />
<br />
For a smooth proper variety over a field of characteristic<br />
zero, the Hodge-to-de Rham spectral sequence (relating the cohomology<br />
of differential forms to de Rham cohomology) is well-known to<br />
degenerate, via Hodge theory. A "noncommutative" version of this<br />
theorem has been proved by Kaledin for smooth proper dg categories<br />
over a field of characteristic zero, based on the technique of<br />
reduction mod p. I will describe a short proof of this theorem using<br />
the theory of topological Hochschild homology, which provides a<br />
canonical one-parameter deformation of Hochschild homology in<br />
characteristic p.<br />
<br />
===Andrei Caldararu===<br />
'''Categorical Gromov-Witten invariants beyond genus 1'''<br />
<br />
In a seminal work from 2005 Kevin Costello defined numerical invariants associated to a <br />
Calabi-Yau A-infinity category. These invariants are supposed to generalize the classical<br />
Gromov-Witten invariants (counting curves in a target symplectic manifold) when the category<br />
is taken to be the Fukaya category. In my talk I shall describe some of the ideas involved in Costello's<br />
approach and recent progress (with Junwu Tu) on extending computations of these invariants<br />
past genus 1.</div>Andreichttps://www.math.wisc.edu/wiki/index.php?title=Algebra_and_Algebraic_Geometry_Seminar_Fall_2018&diff=16004Algebra and Algebraic Geometry Seminar Fall 20182018-09-17T21:35:07Z<p>Andreic: /* Fall 2018 Schedule */</p>
<hr />
<div>The seminar meets on Fridays at 2:25 pm in room B235.<br />
<br />
Here is the schedule for [[Algebra and Algebraic Geometry Seminar Spring 2018 | the previous semester]], [[Algebra and Algebraic Geometry Seminar Spring 2019 | the next semester]], and for [[Algebra and Algebraic Geometry Seminar | this semester]].<br />
<br />
==Algebra and Algebraic Geometry Mailing List==<br />
<br />
<br />
*Please join the [https://admin.lists.wisc.edu/index.php?p=11&l=ags AGS Mailing List] to hear about upcoming seminars, lunches, and other algebraic geometry events in the department (it is possible you must be on a math department computer to use this link).<br />
<br />
== Fall 2018 Schedule ==<br />
<br />
{| cellpadding="8"<br />
!align="left" | date<br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s) <br />
<br />
|-<br />
|September 7<br />
|Daniel Erman<br />
|Big Polynomial Rings<br />
|Local<br />
|-<br />
|September 14<br />
|Akhil Mathew (U Chicago)<br />
|Kaledin's noncommutative degeneration theorem and topological Hochschild homology<br />
|Andrei<br />
|-<br />
|September 21<br />
|Andrei Caldararu<br />
|Categorical Gromov-Witten invariants beyond genus 1<br />
|Local<br />
|-<br />
|September 28<br />
|Mark Walker (Nebraska)<br />
|TBD<br />
|Michael and Daniel<br />
|-<br />
|October 5<br />
|-<br />
|-<br />
|-<br />
|-<br />
|October 12<br />
|Jose Rodriguez (Wisconsin)<br />
|TBD<br />
|Local<br />
|-<br />
|October 19<br />
|Oleksandr Tsymbaliuk (Yale)<br />
|TBD<br />
|Paul Terwilliger<br />
|-<br />
|October 26<br />
|<br />
|<br />
|<br />
|-<br />
|November 2<br />
|Behrouz Taji (Notre Dame)<br />
|TBD<br />
|Botong Wang<br />
|-<br />
|November 9<br />
|Juliette Bruce<br />
|TBD<br />
|Local<br />
|-<br />
|November 16<br />
|Wanlin Li<br />
|TBD<br />
|Local<br />
|-<br />
|November 23<br />
|Thanksgiving<br />
|No Seminar<br />
|<br />
|-<br />
|November 30<br />
|[http://www-personal.umich.edu/~grifo/ Eloísa Grifo] (Michigan)<br />
|TBD<br />
|Daniel<br />
|-<br />
|December 7<br />
|Michael Brown<br />
|TBD<br />
|Local<br />
|-<br />
|December 14<br />
|John Wiltshire-Gordon<br />
|TBD<br />
|Local<br />
|-<br />
|}<br />
<br />
== Abstracts ==<br />
<br />
===Akhil Mathew===<br />
<br />
'''Title: Kaledin's noncommutative degeneration theorem and topological Hochschild homology'''<br />
<br />
For a smooth proper variety over a field of characteristic<br />
zero, the Hodge-to-de Rham spectral sequence (relating the cohomology<br />
of differential forms to de Rham cohomology) is well-known to<br />
degenerate, via Hodge theory. A "noncommutative" version of this<br />
theorem has been proved by Kaledin for smooth proper dg categories<br />
over a field of characteristic zero, based on the technique of<br />
reduction mod p. I will describe a short proof of this theorem using<br />
the theory of topological Hochschild homology, which provides a<br />
canonical one-parameter deformation of Hochschild homology in<br />
characteristic p.</div>Andreichttps://www.math.wisc.edu/wiki/index.php?title=Algebra_and_Algebraic_Geometry_Seminar_Fall_2018&diff=15924Algebra and Algebraic Geometry Seminar Fall 20182018-09-07T19:22:27Z<p>Andreic: </p>
<hr />
<div>The seminar meets on Fridays at 2:25 pm in room B235.<br />
<br />
Here is the schedule for [[Algebra and Algebraic Geometry Seminar Spring 2018 | the previous semester]].<br />
<!--, [[Algebraic Geometry Seminar Spring 2018 | the next semester]], and for [[Algebraic Geometry Seminar | this semester]]. --><br />
<br />
==Algebra and Algebraic Geometry Mailing List==<br />
<br />
<br />
*Please join the [https://admin.lists.wisc.edu/index.php?p=11&l=ags AGS Mailing List] to hear about upcoming seminars, lunches, and other algebraic geometry events in the department (it is possible you must be on a math department computer to use this link).<br />
<br />
== Fall 2018 Schedule ==<br />
<br />
{| cellpadding="8"<br />
!align="left" | date<br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s) <br />
<br />
|-<br />
|September 7<br />
|Daniel Erman<br />
|Big Polynomial Rings<br />
|Local<br />
|-<br />
|September 14<br />
|Akhil Mathew (U Chicago)<br />
|TBA<br />
|Andrei<br />
|-<br />
|September 21<br />
|Andrei Caldararu<br />
|TBA<br />
|Local<br />
|-<br />
|September 28<br />
|Mark Walker (Nebraska)<br />
|TBD<br />
|Michael and Daniel<br />
|-<br />
|October 5<br />
|-<br />
|-<br />
|-<br />
|-<br />
|October 12<br />
|Jose Rodriguez (Wisconsin)<br />
|TBD<br />
|Local<br />
|-<br />
|October 19<br />
|Oleksandr Tsymbaliuk (Yale)<br />
|TBD<br />
|Paul Terwilliger<br />
|-<br />
|October 26<br />
|<br />
|<br />
|<br />
|-<br />
|November 2<br />
|Behrouz Taji (Notre Dame)<br />
|TBD<br />
|Botong Wang<br />
|-<br />
|November 9<br />
|-TBD<br />
|-TBD<br />
|-TBD<br />
|-<br />
|November 16<br />
|Wanlin Li<br />
|TBD<br />
|Local<br />
|-<br />
|November 23<br />
|Thanksgiving<br />
|No Seminar<br />
|<br />
|-<br />
|November 30<br />
|John Wiltshire-Gordon<br />
|TBD<br />
|Local<br />
|-<br />
|December 7<br />
|Michael Brown<br />
|TBD<br />
|Local<br />
|-<br />
|December 14<br />
|-TBD<br />
|-TBD<br />
|-TBD<br />
|-<br />
|}<br />
<br />
== Abstracts ==<br />
<br />
===Akhil Mathew===<br />
<br />
'''Title: Kaledin's noncommutative degeneration theorem and topological<br />
Hochschild homology'''<br />
<br />
For a smooth proper variety over a field of characteristic<br />
zero, the Hodge-to-de Rham spectral sequence (relating the cohomology<br />
of differential forms to de Rham cohomology) is well-known to<br />
degenerate, via Hodge theory. A "noncommutative" version of this<br />
theorem has been proved by Kaledin for smooth proper dg categories<br />
over a field of characteristic zero, based on the technique of<br />
reduction mod p. I will describe a short proof of this theorem using<br />
the theory of topological Hochschild homology, which provides a<br />
canonical one-parameter deformation of Hochschild homology in<br />
characteristic p.</div>Andreichttps://www.math.wisc.edu/wiki/index.php?title=Algebra_and_Algebraic_Geometry_Seminar_Fall_2018&diff=15923Algebra and Algebraic Geometry Seminar Fall 20182018-09-07T19:20:22Z<p>Andreic: </p>
<hr />
<div>The seminar meets on Fridays at 2:25 pm in room B235.<br />
<br />
Here is the schedule for [[Algebra and Algebraic Geometry Seminar Spring 2018 | the previous semester]].<br />
<!--, [[Algebraic Geometry Seminar Spring 2018 | the next semester]], and for [[Algebraic Geometry Seminar | this semester]]. --><br />
<br />
==Algebra and Algebraic Geometry Mailing List==<br />
<br />
<br />
*Please join the [https://admin.lists.wisc.edu/index.php?p=11&l=ags AGS Mailing List] to hear about upcoming seminars, lunches, and other algebraic geometry events in the department (it is possible you must be on a math department computer to use this link).<br />
<br />
== Fall 2018 Schedule ==<br />
<br />
{| cellpadding="8"<br />
!align="left" | date<br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s) <br />
<br />
|-<br />
|September 7<br />
|Daniel Erman<br />
|Big Polynomial Rings<br />
|Local<br />
|-<br />
|September 14<br />
|Akhil Mathew (U Chicago)<br />
|TBA<br />
|Andrei<br />
|-<br />
|September 21<br />
|Andrei Caldararu<br />
|TBA<br />
|Local<br />
|-<br />
|September 28<br />
|Mark Walker (Nebraska)<br />
|TBD<br />
|Michael and Daniel<br />
|-<br />
|October 5<br />
|-<br />
|-<br />
|-<br />
|-<br />
|October 12<br />
|Jose Rodriguez (Wisconsin)<br />
|TBD<br />
|Local<br />
|-<br />
|October 19<br />
|Oleksandr Tsymbaliuk (Yale)<br />
|TBD<br />
|Paul Terwilliger<br />
|-<br />
|October 26<br />
|<br />
|<br />
|<br />
|-<br />
|November 2<br />
|Behrouz Taji (Notre Dame)<br />
|TBD<br />
|Botong Wang<br />
|-<br />
|November 9<br />
|-TBD<br />
|-TBD<br />
|-TBD<br />
|-<br />
|November 16<br />
|Wanlin Li<br />
|TBD<br />
|Local<br />
|-<br />
|November 23<br />
|Thanksgiving<br />
|No Seminar<br />
|<br />
|-<br />
|November 30<br />
|John Wiltshire-Gordon<br />
|TBD<br />
|Local<br />
|-<br />
|December 7<br />
|Michael Brown<br />
|TBD<br />
|Local<br />
|-<br />
|December 14<br />
|-TBD<br />
|-TBD<br />
|-TBD<br />
|-<br />
|}<br />
<br />
== Abstracts ==<br />
<br />
===Akhil Mathew===</div>Andreichttps://www.math.wisc.edu/wiki/index.php?title=Algebra_and_Algebraic_Geometry_Seminar_Fall_2018&diff=15918Algebra and Algebraic Geometry Seminar Fall 20182018-09-07T17:35:07Z<p>Andreic: /* Fall 2018 Schedule */</p>
<hr />
<div>The seminar meets on Fridays at 2:25 pm in room B235.<br />
<br />
Here is the schedule for [[Algebra and Algebraic Geometry Seminar Spring 2018 | the previous semester]].<br />
<!--, [[Algebraic Geometry Seminar Spring 2018 | the next semester]], and for [[Algebraic Geometry Seminar | this semester]]. --><br />
<br />
==Algebra and Algebraic Geometry Mailing List==<br />
<br />
<br />
*Please join the [https://admin.lists.wisc.edu/index.php?p=11&l=ags AGS Mailing List] to hear about upcoming seminars, lunches, and other algebraic geometry events in the department (it is possible you must be on a math department computer to use this link).<br />
<br />
== Fall 2018 Schedule ==<br />
<br />
{| cellpadding="8"<br />
!align="left" | date<br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s) <br />
<br />
|-<br />
|September 7<br />
|Daniel Erman<br />
|Big Polynomial Rings<br />
|Local<br />
|-<br />
|September 14<br />
|Akhil Mathew (U Chicago)<br />
|TBA<br />
|Andrei<br />
|-<br />
|September 21<br />
|Andrei Caldararu<br />
|TBA<br />
|Local<br />
|-<br />
|September 28<br />
|Mark Walker (Nebraska)<br />
|TBD<br />
|Michael and Daniel<br />
|-<br />
|October 5<br />
|-<br />
|-<br />
|-<br />
|-<br />
|October 12<br />
|Jose Rodriguez (Wisconsin)<br />
|TBD<br />
|Local<br />
|-<br />
|October 19<br />
|Oleksandr Tsymbaliuk (Yale)<br />
|TBD<br />
|Paul Terwilliger<br />
|-<br />
|October 26<br />
|<br />
|<br />
|<br />
|-<br />
|November 2<br />
|Behrouz Taji (Notre Dame)<br />
|TBD<br />
|Botong Wang<br />
|-<br />
|November 9<br />
|-TBD<br />
|-TBD<br />
|-TBD<br />
|-<br />
|November 16<br />
|Wanlin Li<br />
|TBD<br />
|Local<br />
|-<br />
|November 23<br />
|Thanksgiving<br />
|No Seminar<br />
|<br />
|-<br />
|November 30<br />
|John Wiltshire-Gordon<br />
|TBD<br />
|Local<br />
|-<br />
|December 7<br />
|Michael Brown<br />
|TBD<br />
|Local<br />
|-<br />
|December 14<br />
|-TBD<br />
|-TBD<br />
|-TBD<br />
|-<br />
|}<br />
<br />
== Abstracts ==</div>Andreichttps://www.math.wisc.edu/wiki/index.php?title=Algebra_and_Algebraic_Geometry_Seminar_Fall_2018&diff=15906Algebra and Algebraic Geometry Seminar Fall 20182018-09-06T21:16:59Z<p>Andreic: /* Fall 2018 Schedule */</p>
<hr />
<div>The seminar meets on Fridays at 2:25 pm in room B235.<br />
<br />
Here is the schedule for [[Algebra and Algebraic Geometry Seminar Spring 2018 | the previous semester]].<br />
<!--, [[Algebraic Geometry Seminar Spring 2018 | the next semester]], and for [[Algebraic Geometry Seminar | this semester]]. --><br />
<br />
==Algebra and Algebraic Geometry Mailing List==<br />
<br />
<br />
*Please join the [https://admin.lists.wisc.edu/index.php?p=11&l=ags AGS Mailing List] to hear about upcoming seminars, lunches, and other algebraic geometry events in the department (it is possible you must be on a math department computer to use this link).<br />
<br />
== Fall 2018 Schedule ==<br />
<br />
{| cellpadding="8"<br />
!align="left" | date<br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s) <br />
<br />
|-<br />
|September 7<br />
|Daniel Erman<br />
|Big Polynomial Rings<br />
|Local<br />
|-<br />
|September 14<br />
|<br />
|<br />
|<br />
|-<br />
|September 21<br />
|Andrei Caldararu<br />
|TBA<br />
|Local<br />
|-<br />
|September 28<br />
|Mark Walker (Nebraska)<br />
|TBD<br />
|Michael and Daniel<br />
|-<br />
|October 5<br />
|-<br />
|-<br />
|-<br />
|-<br />
|October 12<br />
|Jose Rodriguez (Wisconsin)<br />
|TBD<br />
|Local<br />
|-<br />
|October 19<br />
|Oleksandr Tsymbaliuk (Yale)<br />
|TBD<br />
|Paul Terwilliger<br />
|-<br />
|October 26<br />
|<br />
|<br />
|<br />
|-<br />
|November 2<br />
|Behrouz Taji (Notre Dame)<br />
|TBD<br />
|Botong Wang<br />
|-<br />
|November 9<br />
|-TBD<br />
|-TBD<br />
|-TBD<br />
|-<br />
|November 16<br />
|Wanlin Li<br />
|TBD<br />
|Local<br />
|-<br />
|November 23<br />
|Thanksgiving<br />
|No Seminar<br />
|<br />
|-<br />
|November 30<br />
|John Wiltshire-Gordon<br />
|TBD<br />
|Local<br />
|-<br />
|December 7<br />
|Michael Brown<br />
|TBD<br />
|Local<br />
|-<br />
|December 14<br />
|-TBD<br />
|-TBD<br />
|-TBD<br />
|-<br />
|}<br />
<br />
== Abstracts ==</div>Andreichttps://www.math.wisc.edu/wiki/index.php?title=Algebra_and_Algebraic_Geometry_Seminar&diff=15817Algebra and Algebraic Geometry Seminar2018-09-01T20:49:36Z<p>Andreic: Redirected page to Algebra and Algebraic Geometry Seminar Fall 2018</p>
<hr />
<div>#REDIRECT [[Algebra and Algebraic Geometry Seminar Fall 2018]]</div>Andreichttps://www.math.wisc.edu/wiki/index.php?title=Algebra_and_Algebraic_Geometry_Seminar_Fall_2018&diff=15742Algebra and Algebraic Geometry Seminar Fall 20182018-08-22T12:20:14Z<p>Andreic: </p>
<hr />
<div>The seminar meets on Fridays at 2:25 pm in room B235.<br />
<br />
Here is the schedule for [[Algebra and Algebraic Geometry Seminar Spring 2018 | the previous semester]].<br />
<!--, [[Algebraic Geometry Seminar Spring 2018 | the next semester]], and for [[Algebraic Geometry Seminar | this semester]]. --><br />
<br />
==Algebra and Algebraic Geometry Mailing List==<br />
<br />
<br />
*Please join the [https://admin.lists.wisc.edu/index.php?p=11&l=ags AGS Mailing List] to hear about upcoming seminars, lunches, and other algebraic geometry events in the department (it is possible you must be on a math department computer to use this link).<br />
<br />
== Spring 2018 Schedule ==<br />
<br />
{| cellpadding="8"<br />
!align="left" | date<br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s) <br />
<br />
|- }<br />
<br />
== Abstracts ==</div>Andreic