{
  "id": "math/0610732",
  "version": "v1",
  "published": "2006-10-24T16:14:34.000Z",
  "updated": "2006-10-24T16:14:34.000Z",
  "title": "On squares in Lucas sequences",
  "authors": [
    "A. Bremner",
    "N. Tzanakis"
  ],
  "comment": "11 pages. To appear in Journal of Number Theory",
  "categories": [
    "math.NT"
  ],
  "abstract": "Let P and Q be non-zero 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 question of when U_n(P,Q) can be a perfect square has generated interest in the literature. We show that for n=2,...,7, U_n is a square for infinitely many pairs (P,Q) with gcd(P,Q)=1; further, for n=8,...,12, the only non-degenerate sequences where gcd(P,Q)=1 and U_n(P,Q)=square, are given by U_8(1,-4)=21^2, U_8(4,-17)=620^2, and U_12(1,-1)=12^2.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2006-10-24T16:14:34.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11B37",
      "11D41",
      "11G05",
      "11G30"
    ],
    "keywords": [
      "lucas sequence",
      "non-zero integers",
      "non-degenerate sequences",
      "perfect square"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 11,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2006math.....10732B"
    }
  }
}