{
  "id": "math/0405306",
  "version": "v1",
  "published": "2004-05-15T07:12:35.000Z",
  "updated": "2004-05-15T07:12:35.000Z",
  "title": "Lucas sequences whose 12th or 9th term is a square",
  "authors": [
    "Andrew Bremner",
    "Nikos Tzanakis"
  ],
  "comment": "13 pages",
  "categories": [
    "math.NT"
  ],
  "abstract": "Let P and Q be non-zero relatively prime integers. The Lucas sequence {U_n(P,Q) is defined by U_0=0, U_1=1, U_n = P U_{n-1}-Q U_{n-2} for n>1. The sequence {U_n(1,-1)} is the familiar Fibonacci sequence, and it was proved by Cohn that the only perfect square greater than 1 in this sequence is $U_{12}=144$. The question arises, for which parameters P, Q, can U_n(P,Q) be a perfect square? In this paper, we complete recent results of Ribenboim and MacDaniel. Under the only restriction GCD(P,Q)=1 we determine all Lucas sequences {U_n(P,Q)} with U_{12}= square. It turns out that the Fibonacci sequence provides the only example. Moreover, we also determine all Lucas sequences {U_n(P,Q) with U_9= square.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2004-05-15T07:12:35.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11B39",
      "11G07",
      "11G05",
      "11Y50"
    ],
    "keywords": [
      "lucas sequence",
      "9th term",
      "non-zero relatively prime integers",
      "perfect square greater",
      "familiar fibonacci sequence"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 13,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2004math......5306B"
    }
  }
}