The four-squares theorem

Abstract

Abstract We are now in a position to use the ring of Gaussian integers to prove one of the most famous theorems in number theory: every positive integer is the sum of four squares. The material of this chapter is a bit tricky and is not used later, so that the reader may omit it or read it superficially without worrying about the details. Nevertheless, we recommend that you look at Comments 4.7 and the Exercises. Let us set the scene by considering some typical problems of classical number theory. Because for the time being we are in number-theory mode, when we say ‘square’ we mean ‘square of non-negative integer’ and similarly for cubes, etc. There has been a great deal of interest in questions about whether every positive integer can be represented as the sum of such-and-such a number of special integers; or which integers can be represented in this way; or can every large enough integer be represented in this way. Here are some examples.