{
  "id": "math/0408371",
  "version": "v2",
  "published": "2004-08-26T09:53:29.000Z",
  "updated": "2004-08-27T07:02:12.000Z",
  "title": "Lucas sequences whose 8th term is a square",
  "authors": [
    "Andrew Bremner",
    "Nikos Tzanakis"
  ],
  "comment": "21 pages + appendix of 23 pages with computational information",
  "categories": [
    "math.NT"
  ],
  "abstract": "Let P and Q be non-zero integers. The Lucas sequence U_n(P,Q), n=0,1,2,... is defined by U_0=0, U_1=1, U_n= P U_{n-1}-Q U_{n-2} for n>1. For each positive integer n<8 we describe all Lucas sequences with (P,Q)=1 having the property that U_n(P,Q) is a perfect square. The arguments are elementary. The main part of the paper is devoted to finding all Lucas sequences such that U_8(P,Q) is a perfect square. This reduces to a number of problems of similar type, namely, finding all points on an elliptic curve defined over a quartic number field subject to a ``Q-rationality'' condition on the X-coordinate. This is achieved by p-adic computations (for a suitable prime p) using the formal group of the elliptic curve.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2004-08-27T07:02:12.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11B39",
      "11G05",
      "11D25"
    ],
    "keywords": [
      "lucas sequence",
      "8th term",
      "perfect square",
      "elliptic curve",
      "quartic number field subject"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 21,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2004math......8371B"
    }
  }
}