Representations of Integers as Sums of 32 Squares

Abstract

In this paper, we derive a new explicit formula for r32(n), where rk(n) is the number of representations of n as a sum of k squares. For a fixed integer k, our method can be used to derive explicit fomulas for r8k(n). We conclude the paper with various conjectures that lead to explicit formulas for r2k(n), for any fixed positive integer k > 4.