{
  "id": "0909.1663",
  "version": "v4",
  "published": "2009-09-09T09:59:59.000Z",
  "updated": "2013-01-24T10:32:30.000Z",
  "title": "Five squares in arithmetic progression over quadratic fields",
  "authors": [
    "Enrique González-Jiménez",
    "Xavier Xarles"
  ],
  "comment": "To appear in Revista Matem\\'atica Iberoamericana",
  "journal": "Revista Matem\\'atica Iberoamericana 29, no. 4, 1211-1238 (2013)",
  "doi": "10.4171/RMI/754",
  "categories": [
    "math.NT"
  ],
  "abstract": "We give several criteria to show over which quadratic number fields Q(sqrt{D}) there should exists a non-constant arithmetic progressions of five squares. This is done by translating the problem to determining when some genus five curves C_D defined over Q have rational points, and then using a Mordell-Weil sieve argument among others. Using a elliptic Chabauty-like method, we prove that the only non-constant arithmetic progressions of five squares over Q(sqrt{409}), up to equivalence, is 7^2, 13^2, 17^2, 409, 23^2. Furthermore, we give an algorithm that allow to construct all the non-constant arithmetic progressions of five squares over all quadratic fields. Finally, we state several problems and conjectures related to this problem.",
  "revisions": [
    {
      "version": "v4",
      "updated": "2013-01-24T10:32:30.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11G30",
      "11B25",
      "11D45",
      "14H25"
    ],
    "keywords": [
      "quadratic fields",
      "non-constant arithmetic progressions",
      "mordell-weil sieve argument",
      "quadratic number fields",
      "rational points"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2009arXiv0909.1663G"
    }
  }
}