All square: a surprising, far-reaching overhaul for theories about quadratic expressions

Science News, March 11, 2006 by Ivars Peterson

Start with the square numbers 1, 4, 9, 16, 25, 36, and so on. Pick any other number and you can express it as a sum of squares. For example, 10 = 1 1 4 4 and 30 = 1 4 9 16. In 1770, French mathematician Joseph-Louis Lagrange proved that every positive integer is either a square itself or the sum of two, three, or four squares. No more than four squares, [x.sup.2] [y.sup.2] [z.sup.2] [t.sup.2], are ever needed to express any number, no matter how large.

Given Lagrange's result, number theorists asked whether there are other such expressions, called quadratic forms, that also repre sent all positive integers. In 1916, Indian mathematician Srinivasa Ramanujan uncovered 53 such expressions. He showed, for instance, that every number could be written as a square plus twice a square plus three times a square plus five times a square.

Ramanujan's discovery led mathematicians to tackle the general question: How can you predict when a quadratic form represents all positive integers?

Now, Manjul Bhargava of Princeton University and Jonathan P. Hanke of Duke University in Durham, N.C., have demonstrated that this question has a simple, surprising answer. Bhargava described the findings--a set of theorems--at the International Conference on Number Theory and Mathematical Physics, held last December at SASTRA University in Kumbakonam, India.

The work has turned up "many amazing results," says Ken Ono of the University of Wisconsin-Madison. "They are the ultimate theorems in the theory of representations of integers as sums of squares."

COMPOSITION LAWS Bhargava has had a strong interest in math since childhood (see box p. 153). His mathematical research dating back to his graduate school years at Princeton contributed to the recent findings.

In his 2001 Ph.D. thesis, Bhargava proved several important theorems in a branch of number theory that grew out of work done by German mathematician Karl Friedrich Gauss in 1801. It focused on the polynomial expressions called quadratic forms, such as [ax.sup.2] bxy [cy.sup.2], in which each term has a variable with an exponent of 2 or is the product of two variables.

Gauss had developed a method for combining two quadratic equations ([ax.sup.2] bxy [cy.sup.2] = O) in a way that differs from normal addition. Such a method is known as a composition law. Gauss' composition law offered a new way of thinking about relationships among numbers and led to the development of the mathematical field now known as algebraic number theory.

In his graduate work, Bhargava discovered 13 composition laws and developed a mathematical framework for explaining them.

"Bhargava's work is marked by extreme ingenuity and depth," says Krishnaswami Alladi of the University of Florida in Gainesville. Bhargava's unexpected discovery of new composition laws created an exciting line of research on a topic that had seen little activity since the time of Gauss.

For these novel and elegant formulations, Bhargara earned not only his doctorate but also a prestigious fellowship at the Clay Mathematics Institute in Cambridge, Mass. In 2003, at age 28, he joined the Princeton faculty as a full professor, one of the youngest scholars ever to achieve this rank.

MAGIC 15 In the meantime, Bhargava had become intrigued by the question of which quadratic forms represent all positive integers. Mathematicians call such expressions universal quadratic forms.

Early last century, Ramanujan focused on quadratic expressions of the form [ax.sup.2] [by.sup.2] [cz.sup.2] [dt.sup.2], where a, b, c, and d are integers. He found the 53 universal quadratics in that group. For example, varying the values of x, y, z, and t in the quadratic form [x.sup.2] 2[y.sup.2] 5[z.sup.2] 10[t.sup.2] can generate all positive integers. To get 14, set x = 1, y = 2, z = 1, t = 0; to get 32, set x = 0, y = 1, z = 2, and t = 1.

Mathematicians wanted to know how many more universal quadratics exist. They therefore needed a simple test to determine whether a given form represents all positive integers.

In 1993, John H. Conway of Princeton and his student William Schneeberger announced a startling result that applies to a subset of quadratic forms, those that can be defined by a specific type of matrix. To decide whether a form within that group is universal, they needed only to check whether it can represent each of nine integers: 1, 2, 3, 5, 6, 7, 10, 14, and 15. If it does, then that particular quadratic form represents all the positive integers.

This result came to be known as the "15 theorem." It serves as a filter, separating a certain group of quadratic forms into universal and nonuniversal expressions.

The universal quadratics previously recognized by Lagrange and Ramanujan belong to the matrix-defined subset considered by Conway and Schneeberger. So, the 15 theorem also provides a rapid proof of Lagrange's four-square theorem.

Conway and Schneeberger never published their lengthy and intricate proof of the 15 theorem. They proceeded, however, to try to extend it to a much wider group of quadratic forms, described as integer valued. This group includes expressions such as 3[x.sup.2] xy 5[y.sup.2] 6[z.sup.2] [t.sup.2].