Large Sums of 4k Squares: A Polynomial Approach (pp. 133-152)
Authors: Aycse Alaca, Saban Alaca and Kenneth S. Williams
Abstract: Let k and n be positive integers. Let sk(n)denote the number of representations of n as the sum of k squares. Ramanujan , [18, p. 159] gave without proof a formula for sk(n) when k is even. Mordell  used modular forms to give the first proof of Ramanujanís formula. In 2001 Cooper  used Ramanujanís 1ψ1 summation formula and Jacobian elliptic functions and their derivatives to give a proof. It is our purpose to show that when k is a multiple of 4, Ramanujanís formula can be proved in an entirely elementary way using the properties of a certain class of polynomials. The values of sk(n) are determined explicitly for k = 4,8,12,16,20,24,28,32,36,40,44 and 48.