Analogues of Jacobi's two-square theorem: an informal account

Abstract

Jacobi's two-square theorem states that the number of representations of a positive integer k as a sum of two squares, counting order and sign, is 4 times the surplus of positive divisors of k congruent to 1 modulo 4 over those congruent to 3 modulo 4. In this paper we give numerous identities, each of which yields an analogue of Jacobi's result. Our identities are drawn from a much larger list, and involve polygonal numbers. The formula for the nth k-gonal number is