{
  "id": "1005.1528",
  "version": "v1",
  "published": "2010-05-10T13:01:35.000Z",
  "updated": "2010-05-10T13:01:35.000Z",
  "title": "The Diophantine equation $x^4\\pm y^4=iz^2$ in Gaussian integers",
  "authors": [
    "Filip Najman"
  ],
  "comment": "5 pages, to appear in Amer. Math. Monthly",
  "journal": "Amer. Math. Monthly 117 (2010), 637-641",
  "categories": [
    "math.NT"
  ],
  "abstract": "In this note we find all the solutions of the Diophantine equation $x^4\\pm y^4=iz^2$ using elliptic curves over $\\mathbb Q(i)$. Also, using the same method we give a new proof of Hilbert's result that the equation $x^4\\pm y^4=z^2$ has only trivial solutions in Gaussian integers.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2010-05-10T13:01:35.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "gaussian integers",
      "diophantine equation",
      "hilberts result",
      "trivial solutions"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 5,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2010arXiv1005.1528N"
    }
  }
}