Science News, June 16, 2001; Vol. 159, No. 24
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Surprisingly Square
Mathematicians take a fresh look at expressing
numbers as the sums of squares
By Ivars Peterson
For many decades, the study of the sums of squares was a stagnant
backwater of mathematical research. This state of affairs changed
unexpectedly in 1996 when mathematician Stephen C. Milne of Ohio
State University in Columbus unveiled powerful new formulas for
enumerating representations of numbers as the sums of squares.
Milne's discoveries "came as a great surprise," says Ken Ono of the
University of Wisconsin–Madison. "It's amazing that he found those
relations."
Many mathematicians greeted Milne's startling results with
skepticism, however. Milne's published announcement provided only a
sketchy outline of his work. Moreover, the formulas he had obtained
were exceedingly complicated, making them difficult to understand and
apply.
Now, those initial doubts have evaporated. Details of Milne's
groundbreaking research will be published next year as a 125-page
paper in a special issue of the Ramanujan Journal.
In the meantime, Ono and other mathematicians have used a different
mathematical approach to provide much shorter proofs of some of
Milne's main results and to furnish simpler formulas for counting
representations of numbers as the sums of squares.
"Without Milne's pioneering effort, many of us would not have been
thinking about the problem," Ono says.
The study of the sums of squares has a lengthy history, and it
remains an important area of research in pure mathematics, says
George E. Andrews of Pennsylvania State University in University Park.
Nearly 2,000 years ago, for instance, Diophantus of Alexandria
observed in his book Arithmetica that 65 can be written in two
different ways as the sum of two squares: 4^2 + 7^2 and 8^2 + 1^2 . He
went on to detail a variety of relationships involving squares of
integers.
Modern efforts have focused on finding formulas that give the number
of different ways in which an integer can be represented as the sum
of a given number of squares.
Consider the sequence of squares of whole numbers: 0, 1, 4, 9, 16,
and so forth. As the squares get larger, the gaps between consecutive
squares get wider. Clearly, most integers are not squares of whole
numbers.
Many integers can be written as the sum of two squares: 8 = 4 + 4; 10
= 9 + 1; 13 = 9 + 4; and so on. Other numbers can't be expressed as
the sum of just two squares, however. To get a sum that equals 6, the
only squares available are 4 and 1, and that won't do the job.
Instead, it takes the sum of three squares: 4 + 1 + 1.
Indeed, most positive integers can be written as the sum of three
squares. For instance, 11 = 9 + 1 + 1 and 12 = 4 + 4 + 4.
On the other hand, 7 is an example of an integer that can't be
written as the sum of three squares. It takes four squares: 7 = 4 + 1
+ 1 + 1.
Do you ever need more than four squares to express an integer? In
1770, French mathematician Joseph-Louis Lagrange proved what
Diophantus, Pierre de Fermat, and others previously assumed: Every
positive integer is either a square itself or the sum of two, three,
or four squares.
Mathematicians also became interested in the number of different ways
in which a given whole number can be expressed as the sum of four or
more squares. In such enumerations, 0 can be included as one of the
square numbers, and negative numbers can be squared.
In 1829, German mathematician Carl Jacobi found formulas that give
the number of representations of an integer as the sum of two, four,
six, or eight squares. To do so, Jacobi worked with mathematical
expressions known as elliptic functions. Such expressions originally
arose in the context of determining the length of a piece of an
ellipse.
Jacobi's formula for representations made up of four squares, for
instance, is simply 8 times the sum of all positive divisors of the
given integer that are not multiples of 4. Suppose the given integer
is 4, which also happens to be a square itself. The positive divisors
of 4 are 1, 2, and 4. Excluding 4, the calculation involves just 1
and 2. Multiplying the sum (1 + 2 = 3) by 8 gives 24 as the number of
different representations of 4 as the sum of four squares (see box,
above).
Similarly, there are 48 representations of 5 as the sum of four
squares, starting with 22 + 12 + 02 + 02. The divisors of 5 are 1 and
5, and neither divisor is a multiple of 4. Applying Jacobi's formula,
the number of representations of 5 in terms of four squares is 8
multiplied by the sum of the divisors (1 + 5 = 6), giving the answer
48.
Jacobi's formulas work for sums of up to eight squares.
Mathematicians then sought to come up with formulas for
representations of numbers using more than eight squares. This effort
tripped over an apparent stumbling block in the 1960s, when Robert A.
Rankin of the University of Glasgow proved a theorem ruling out the
existence of certain types of formulas analogous to the simple ones
found by Jacobi. Rankin's result discouraged other mathematicians
from pursuing the question further.
There was a loophole, however. Rankin's result didn't cover every
possible type of formula, and Milne was one of the very few who
continued the pursuit. Probably no one else believed it possible to
find simple formulas, comments Bruce C. Berndt of the University of
Illinois at Urbana-Champaign.
Returning to the elliptic-function approach pioneered by Jacobi and
combining it with other techniques, Milne eventually discovered new
formulas for the number of representations when more than eight
squares are involved.
Milne's 1996 discovery represented a "startling turnabout," Ono
says.
"He made me believe that simple formulas could exist."
Milne's formulas themselves, however, were hard to fathom and use. To
find simpler versions, mathematicians turned to an alternative
approach that uses mathematical objects known as modular forms.
Mathematicians had developed the theory of modular forms in the early
part of the 20th century to gain deeper insights into number
relationships. A modular form is an abstract, highly symmetric,
impossible-to-visualize mathematical object that encodes
relationships far more complex than those expressed by simple
functions, such as the wavy sine function in trigonometry.
The modular-form approach proved sufficiently powerful that it came
to dominate much of number theory, Andrews says. For example, it
played a central role in the recent proof of Fermat's last theorem by
Andrew Wiles of Princeton University (SN: 10/2/99, p. 221).
In the course of his work on the sums of squares, Milne had proved
conjectures first proposed in 1994 by Victor G. Kac of the
Massachusetts Institute of Technology and Minoru Wakimoto of Kyushu
University in Fukuoka, Japan. The conjectures concerned the problem
of writing an integer as the sum of three triangular numbers. This
challenge is closely connected to the problem of writing an integer
as the sum of three squares. A triangular number has the form k (k +
1)/2, for k = 1, 2, 3, . . ., so the triangular numbers are 1, 3, 6,
10, and so on (see box, above).
Last year, working independently, number theorist Don Zagier of the
Max Planck Institute for Mathematics in Bonn, Germany, used a
modular-forms approach to provide a significantly shorter proof of
the Kac-Wakimoto conjectures. Zagier's version appeared in the
September-November 2000 Mathematical Research Letters.
Zagier's method "involves an elegant and surprisingly simple
argument," Ono notes.
Earlier this year, Ono extended Zagier's results to derive new
formulas for representations of sums of squares that are considerably
simpler than those of Milne. Ono "gives cleaner formulas and far
shorter proofs," Berndt says. "But he owes a debt to Milne, for Ono
would not have discovered his theorems if it had not been for Milne's
work."
To tackle questions concerning sums of squares, mathematicians now
have two distinctly different approaches—the one rooted in the theory
of elliptic functions and the other in the theory of modular
forms."It will take quite a while to see which method will open up
further new results and not just give new proofs," remarks
mathematician Richard Askey, also of the University of
Wisconsin-Madison. So far, the modular-forms method has only
confirmed Milne's work.
"My hunch is that both methods will lead to surprises, but probably
in different ways," says Askey.
The two approaches to the study of sums of squares "are greatly
enriching both areas of mathematics," Milne suggests.
"Now, we have an interesting situation where there are many more
questions," he says. Why do the two seemingly unrelated approaches
give the same results? "In particular, what is the exact nature of
the beautiful relations between [the methods]?" he asks.
The recent ventures of Milne, Zagier, and Ono could very well
represent just the first of many productive forays into a venerable
area where mathematicians had made little progress in recent decades.
References and Sources
References:
Milne, S.C. In press. Infinite families of exact sums of squares
formulas, Jacobi elliptic functions, continued fractions, and Schur
functions. Ramanujan Journal. Available as a preprint at
http://xxx.lanl.gov/abs/math.NT/0008068.
Ono, K. Preprint. Representations of integers as sums of squares.
Zagier, D. 2000. A proof of the Kac-Wakimoto affine denominator
formula for the strange series. Mathematical Research Letters
7(September-November):597.
Further Readings:
Kac, V.G., and M. Wakimoto. 1994. Integrable highest weight modules
over affine superalgebras and Appell's function. In Progress in
Mathematics, eds. J.-L. Brylinski, et al. Boston, Mass.: Birkhauser.
Milne, S.C. 1996. New infinite families of exact sums of squares
formulas, Jacobi elliptic functions, and Ramanujan's tau function.
Proceedings of the National Academy of Sciences 93(Dec. 24):15004.
Peterson, I. 1999. Curving beyond Fermat. Science News Online.
Available at
http://www.sciencenews.org/sn_arc99/11_20_99/mathland.htm.
______. 1999. Curving beyond Fermat's last theorem. Science News
156(Oct. 2):221.
Sources:
George E. Andrews
Department of Mathematics
Pennsylvania State University
University Park, PA 16802-6402
Richard Askey
Department of Mathematics
University of Wisconsin
Madison, WI 53706
Bruce C. Berndt
Department of Mathematics
University of Illinois
1409 West Green Street
Urbana, IL 61801
Stephen C. Milne
Department of Mathematics
Ohio State University
Columbus, OH 43210
Web site: http://www.math.ohio-state.edu/~milne/
Ken Ono
Department of Mathematics
University of Wisconsin
Madison, WI 53706
Web site: http://www.math.wisc.edu/~ono/
http://www.sciencenews.org/20010616/bob19.asp
From Science News, Vol. 159, No. 24, June 16, 2001, p. 382.
Copyright (c) 2001 Science Service. All rights reserved.
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