Topics in the Theory of Numbers (2nd Ed)
- P Erdos, J Suranyi
- Springer-Verlag New York Inc.
The Four Square Theorem was proved by Lagrange in 1770: every positive integer is the sum of at most four squares of positive integers, i.e. n = A2 + B2 + C2 + D2, A, B, C, D ∈ Z An interesting proof is presented here based on Hurwitz integers, a subset of quarternions which act like integers in four dimensions and have the prime divisor property. It is analogous to Fermat’s Two Square Theorem, which says that positive primes of the form 4k+1 can be written as the sum of the squares of two positive integers. Representing integers as the sum of squares can be considered a special case of Waring’s problem: for every k is there a number g (k) such that every integer is representable by at most g (k) kth powers?