{
  "id": "math/0701252",
  "version": "v1",
  "published": "2007-01-09T15:18:01.000Z",
  "updated": "2007-01-09T15:18:01.000Z",
  "title": "Lucas sequences whose nth term is a square or an almost square",
  "authors": [
    "A. Bremner N. Tzanakis"
  ],
  "comment": "24 pages (double spaced). To appear in Acta Arithmetica",
  "categories": [
    "math.NT"
  ],
  "abstract": "(Below, \\Box means \"perfect square\") Let $P$ and $Q$ be non-zero integers. The Lucas sequence $\\{U_n(P,Q)\\}$ is defined by $U_0=0$, $U_1=1$, $U_n=P U_{n-1}-Q U_{n-2}$, $(n \\geq 2)$. Historically, there has been much interest in when the terms of such sequences are perfect squares (or higher powers). Here, we summarize results on this problem, and investigate for fixed $k$ solutions of $U_n(P,Q)= k\\Box$, $(P,Q)=1$. We show finiteness of the number of solutions, and under certain hypotheses on $n$, describe explicit methods for finding solutions. These involve solving finitely many Thue-Mahler equations. As an illustration of the methods, we find all solutions to $U_n(P,Q)=k\\Box$ where $k=\\pm1,\\pm2$, and $n$ is a power of 2.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2007-01-09T15:18:01.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11D41",
      "11B39",
      "11D59",
      "11G30"
    ],
    "keywords": [
      "lucas sequence",
      "nth term",
      "perfect square",
      "non-zero integers",
      "explicit methods"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 24,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}