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*^{2} + *y*^{2} + *z*^{2} + *t*^{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 represent 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.

Now a Mathematician named Manjul Bhargava from Princeton has shown with a simplified calculation, that altogether there are 204 universal, matrix-defined quadratic forms. Mathematicians previously thought that the tally had been completed in 1948, when Margaret F. Willerding in her doctoral research at St. Louis University painstakingly worked out 178 universal matrix-defined quadratic forms. Bhargava's new enumeration, taking advantage of the shortcut provided by the 15 theorem, shows that Willerding missed 36 universal forms, listed one universal form twice, and included nine forms that are, in fact, not universal.

1## Re: Expressing Positive Integers as the Sum of Squares