{
  "id": "1304.6887",
  "version": "v1",
  "published": "2013-04-25T12:09:04.000Z",
  "updated": "2013-04-25T12:09:04.000Z",
  "title": "Positive Integer Solutions of the Pell Equation $x^{2}-dy^{2}=N,$ $% d\\in \\left\\{k^{2}\\pm 4,\\text{}k^{2}\\pm 1\\right\\} $ and $N\\in \\left\\{\\pm 1,\\pm 4\\right\\}",
  "authors": [
    "Refik Keskin",
    "Merve Güney"
  ],
  "categories": [
    "math.NT"
  ],
  "abstract": "Let $\\ k$ be a natural number and $d=k^{2}\\pm 4$ or $k^{2}\\pm 1$. In this paper, by using continued fraction expansion of $\\sqrt{d},$ we find fundamental solution of the equations $x^{2}-dy^{2}=\\pm 1$ and we get all positive integer solutions of the equations $x^{2}-dy^{2}=\\pm 1$ in terms of generalized Fibonacci and Lucas sequences. Moreover, we find all positive integer solutions of the equations $x^{2}-dy^{2}=\\pm 4$ in terms of generalized Fibonacci and Lucas sequences. Although some of the results are well known, we think our method is elementary and different from the others.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2013-04-25T12:09:04.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11B37",
      "11B39",
      "11B50",
      "11B99",
      "11A55"
    ],
    "keywords": [
      "positive integer solutions",
      "pell equation",
      "lucas sequences",
      "generalized fibonacci",
      "continued fraction expansion"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1304.6887K"
    }
  }
}