{
  "id": "1607.08648",
  "version": "v1",
  "published": "2016-07-28T21:48:24.000Z",
  "updated": "2016-07-28T21:48:24.000Z",
  "title": "Quartic Equations with Trivial Solutions over Gaussian Integers",
  "authors": [
    "Felix Sidokhine"
  ],
  "categories": [
    "math.NT"
  ],
  "abstract": "In our work we study the equations of the form $aX^4+bX^2 Y^2+cY^4=dZ^2$ over Gaussian integers by a method of the resolvents. We study as a new equations $X^4+6X^2 Y^2+Y^4=Z^2$ (Mordell's equation over $\\mathbb{Z}[i]$), $X^4+6(1+i)X^2Y^2+2iY^4=Z^2$ and $X^4\\pm Y^4=(1+ i)Z^2$ and give the new proofs of the known theorems on $X^4+Y^4=Z^2$ (Fermat - Hilbert), $X^4\\pm Y^4=iZ^2$ (Szab\\'o - Najman).",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-07-28T21:48:24.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "gaussian integers",
      "trivial solutions",
      "quartic equations",
      "mordells equation",
      "resolvents"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}