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

## TL;DR

This paper provides a constructive, algorithmic proof of Jacobi's identity for sums of two squares, utilizing combinatorial methods, integer partitions, and matchings on infinite graphs.

## Contribution

It introduces a novel constructive proof and an explicit algorithm for representing integers as sums of two squares based on factorization and combinatorial structures.

## Key findings

- Develops an algorithm transforming factorizations with specific congruences into sums of two squares.
- Uses combinatorial proofs of Jacobi Triple Product and Hirschhorn's proof.
- Frames the problem in terms of integer partitions and matchings on infinite graphs.

## 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 $d \equiv1\mod 4$ plus two bits of data, and whose output is either another factorization $n=d'N'$ and $d' \equiv3\mod 4$ 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.