AbstractsMathematics

A fundamental problem in the theory of numbers is that of finding what numbers are represented in ternary quadratic forms. It was found, in many cases, that the numbers not represented followed some simple law so that, although a certain form failed to represent an infinite number of integers, these integers could be determined by a definite set of formulae. It is easy to see that no number of the form 4^h(8n+7), where h and n are integers >= 0, is a sum of three squares. Legendre proved, conversely, in 1798, that all positive integers not of the form 4^h(8n+7), are sums of three squares, i.e. if N if W <> 4^h(8n+7), h, n >= 0, then N = x^2 + y^2 + z^2 where x, y, z are integers. [...]