Difference between revisions of "Math 567 -- Elementary Number Theory"

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Elementary Number Theory'''
 
Elementary Number Theory'''
  
MWF 1:20-2:10, Van Vleck B119
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TR 9:30-11:00, Van Vleck B119
  
'''Professor:'''  [http://www.math.wisc.edu/~ellenber/ Jordan Ellenberg] (ellenber@math.wisc.edu)
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'''Professor:'''  [http://www.math.wisc.edu/~andreic/ Andrei Caldararu] (andreic@math.wisc.edu)
''Office Hours:'' Weds, 2:30-3:30, Van Vleck 323.  
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''Office Hours:'' Tuesdays, 1:30-3:00, Van Vleck 605.  
  
'''Grader:'''  [http://www.math.wisc.edu/~sjohnson/ Silas Johnson] (snjohnson3@wisc.edu)
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'''Grader:'''  [Shouwei Hui] (shui5@wisc.edu)
''Office Hours:'' Thurs, 1:15-2:15, Van Vleck 522. [http://math.wisc.edu/~sjohnson/567F10 Silas's page], with problem set solutions.
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''Office Hours:'' TBA.
  
 
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://www.sagemath.org/pdf/SageTutorial.pdf useful online tutorial.]  You can download SAGE to your own computer or [http://www.sagenb.org use it online].
 
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://www.sagemath.org/pdf/SageTutorial.pdf useful online tutorial.]  You can download SAGE to your own computer or [http://www.sagenb.org use it online].
  
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. 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.
+
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.
  
  
'''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.
+
'''Course Policies:'''  Homework will be due on Thursdays.  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.
  
 
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.  
 
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.  
  
'''Grading:'''  The grade in Math 567 will be composed of 40% homework, 20% each of three midterms.  The last midterm will be take-home, and will be due on the last day of class.  ''There will be no final exam in Math 567''.
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'''Grading:'''  The grade in Math 567 will be composed of 50% homework, 25% midterm, 25% final.  The final exam date and location will be announced by the University and posted here when available.
  
 
'''Syllabus:'''  
 
'''Syllabus:'''  
 
(This may change as we see what pace works well for the course.  All section numbers refer to Stein's book.)
 
(This may change as we see what pace works well for the course.  All section numbers refer to Stein's book.)
 
   
 
   
* Sep 3-10:  Prime numbers, prime factorizations, Euclidean algorithm and GCD (1.1-1.2)
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* Sep 7 + Sep 11-15:  Prime numbers, prime factorizations, Euclidean algorithm and GCD (1.1-1.2)
* Sep 13-17: The integers mod n, Euler's theorem, the phi function (2.1-2.2)
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* Sep 18-22: The integers mod n, Euler's theorem, the phi function (2.1-2.2)
* Sep 20-24: Modular exponentiation, primality testing, and primitive roots (2.4-2.5)
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* Sep 25-29: Modular exponentiation, primality testing, and primitive roots (2.4-2.5)
* Sep 27-Oct 1:  Public-key cryptography and RSA (3.1-3.4)
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* Oct 2-6:  Public-key cryptography and RSA (3.1-3.4)
* Oct 4 - 8: Algebraic numbers  
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* Oct 9-13: Rabin's algorithm (not in the book); algebraic numbers  
* Oct 6:  ''First midterm exam''
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* Oct 16-20:  Quadratic reciprocity (4.1-4.4)
* Oct 11-15:  Quadratic reciprocity (4.1-4.4)
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* Oct 23-27:  Finite and infinite continued fractions (5.1-5.3)
* Oct 18-22:  Finite and infinite continued fractions (5.1-5.3)
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* Oct 31: ''Midterm exam''
* Oct 25-29: Continued fractions and diophantine approximation (5.4-5.5)
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* Nov 2, Nov 6-10: Continued fractions and diophantine approximation (5.4-5.5)
* Nov 1-5: Diophantine equations I:  Pell's equation and Lagrange's theorem
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* Nov 13-17: Diophantine equations I:  Pell's equation and Lagrange's theorem
* Nov 8-12: Elliptic curves (6.1-6.2)
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* Nov 21 and Nov 28-30: Elliptic curves (6.1-6.2)
* Nov 10: ''Second midterm exam''
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* Dec. 4-8: Applications of elliptic curves (6.3-6.4)
* Nov 15-19: Applications of elliptic curves (6.3-6.4)
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* Dec. 12: advanced topic TBD: maybe additional discussion of cryptographic techniques?
* Nov 22-Dec 3: Diophantine equations II: Fermat, generalized Fermat, and probabilistic methods
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* Dec 6-15: advanced topic TBD: maybe a look at the Sato-Tate conjecture?
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+
  
  
 
'''Homework:'''
 
'''Homework:'''
 
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.
 
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.
 +
 +
<!--
  
 
* '''Sep 13''' (note this is Monday, not Friday!): 1.1, 1.3, 1.5, 1.7 (use SAGE), 1.8, 1.14.
 
* '''Sep 13''' (note this is Monday, not Friday!): 1.1, 1.3, 1.5, 1.7 (use SAGE), 1.8, 1.14.
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OPTIONAL EXTRA:  Can you give an exact formula for the minimal norm of any nonzero complex number of the form a+bz?
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OPTIONAL EXTRA:  Can you give an exact formula for the minimal norm of any nonzero complex number of the form a+bz? -->

Revision as of 10:22, 6 September 2017

MATH 567

Elementary Number Theory

TR 9:30-11:00, Van Vleck B119

Professor: Andrei Caldararu (andreic@math.wisc.edu) Office Hours: Tuesdays, 1:30-3:00, Van Vleck 605.

Grader: [Shouwei Hui] (shui5@wisc.edu) Office Hours: TBA.

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 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, it is available as a free legal .pdf. We will be using the (free, public-domain) mathematical software SAGE, developed largely by Stein, as an integral component of our coursework. There is a useful online tutorial. You can download SAGE to your own computer or use it online.

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.


Course Policies: Homework will be due on Thursdays. 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.

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.

Grading: The grade in Math 567 will be composed of 50% homework, 25% midterm, 25% final. The final exam date and location will be announced by the University and posted here when available.

Syllabus: (This may change as we see what pace works well for the course. All section numbers refer to Stein's book.)

  • Sep 7 + Sep 11-15: Prime numbers, prime factorizations, Euclidean algorithm and GCD (1.1-1.2)
  • Sep 18-22: The integers mod n, Euler's theorem, the phi function (2.1-2.2)
  • Sep 25-29: Modular exponentiation, primality testing, and primitive roots (2.4-2.5)
  • Oct 2-6: Public-key cryptography and RSA (3.1-3.4)
  • Oct 9-13: Rabin's algorithm (not in the book); algebraic numbers
  • Oct 16-20: Quadratic reciprocity (4.1-4.4)
  • Oct 23-27: Finite and infinite continued fractions (5.1-5.3)
  • Oct 31: Midterm exam
  • Nov 2, Nov 6-10: Continued fractions and diophantine approximation (5.4-5.5)
  • Nov 13-17: Diophantine equations I: Pell's equation and Lagrange's theorem
  • Nov 21 and Nov 28-30: Elliptic curves (6.1-6.2)
  • Dec. 4-8: Applications of elliptic curves (6.3-6.4)
  • Dec. 12: advanced topic TBD: maybe additional discussion of cryptographic techniques?


Homework: 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.