Sums of Squares

Abstract

Our main aim in this chapter is to determine which integers can be expressed as the sum of a given number of squares, that is, which have the form where each x _i ε ℤ, for a given k. We shall concentrate mainly on the two most important cases, characterising the sums of two squares, and showing that every non-negative integer is a sum of four squares. We shall adopt two completely different approaches to this problem: the first is mainly algebraic, making use of two number systems, the Gaussian integers and the quaternions; the second approach is geometric, based on the fact that the expression represents the square of the length of the vector (x ₁,…, x _k) in R^k. We shall therefore give two different proofs for several of the main theorems in this chapter. In mathematics, it is often useful to have more than one proof of a result, not because this adds anything to its validity (a single correct proof is enough for this), but rather because the extra proofs may add to our understanding of the result, and may enable us to extend it in different directions.