Surprisingly Square

Mathematicians take a fresh look at expressing numbers as the sums of squares

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.

Two consecutive triangular numbers make a square.

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.

A lengthy history

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: 42 + 72 and 82 + 12 . 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,


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.”

Simpler formulas

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.”

Two approaches

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.

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