A Constructive Proof of Jacobi's Identity for the Sum of Two Squares

2019/07/15 by DeFranco, Mario
#Combinatorics (math.CO) #FOS: Mathematics

paper · doi:10.48550/arxiv.1907.06350

Abstract

We present a constructive proof of Jacobi's identity for the sum of two squares. We present a combinatorial proof of the Jacobi Triple Product and combine with a proof of Hirschhorn to define an algorithm. The input is a factorization n=dN with d ≡1\mod 4 plus two bits of data, and whose output is either another factorization n=d'N' and d' ≡3\mod 4 with two more bits of data, or a pair of integers whose squares sum to n. We phrase this algorithm in terms of integer partitions and matchings on an infinite graph.

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