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Large Sums of 4k Squares: A Polynomial Approach (pp. 133-152) $45.00
Authors:  Aycse Alaca, Saban Alaca and Kenneth S. Williams
Let k and n be positive integers. Let sk(n)denote the number of representations of n as the sum of k squares. Ramanujan [17], [18, p. 159] gave without proof a formula for sk(n) when k is even. Mordell [15] used modular forms to give the first proof of Ramanujanís formula. In 2001 Cooper [6] 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. 

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Large Sums of 4k Squares: A Polynomial Approach (pp. 133-152)