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A Kronecker‐type identity and the representations of a number as a sum of three squares

By considering a limiting case of a Kronecker‐type identity, we obtain an identity found by both Andrews and Crandall. We then use the Andrews–Crandall identity to give a new proof of a formula of Gauss for the representations of a number as a sum of three squares. From the Kronecker‐type identity, we also deduce Gauss' theorem that every positive integer is representable as a sum of three triangular numbers.