{
  "id": "1907.06350",
  "version": "v1",
  "published": "2019-07-15T07:10:41.000Z",
  "updated": "2019-07-15T07:10:41.000Z",
  "title": "A Constructive Proof of Jacobi's Identity for the Sum of Two Squares",
  "authors": [
    "Mario DeFranco"
  ],
  "categories": [
    "math.CO"
  ],
  "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.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-07-15T07:10:41.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "jacobis identity",
      "constructive proof",
      "jacobi triple product",
      "infinite graph",
      "integer partitions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}