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There is a well-known formula due to Jacobi for the number r2(n) of representations of the number n as the sum of two squares. This formula implies that the numbers r2(n) satisfy elegant arithmetic relations. Conversely, these arithmetic properties essentially imply Jacobi's formula. So it is of interest to give direct proofs of these arithmetic relations, and this we do.
We show that Jacobi's two-square theorem is an almost immediate consequence of a famous identity of his, and draw combinatorial conclusions from two identities of Ramanujan.
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