{
  "id": "1905.11322",
  "version": "v1",
  "published": "2019-05-27T16:15:42.000Z",
  "updated": "2019-05-27T16:15:42.000Z",
  "title": "On the $x-$coordinates of Pell equations which are sums of two Padovan numbers",
  "authors": [
    "Mahadi Ddamulira"
  ],
  "comment": "30 pages",
  "categories": [
    "math.NT"
  ],
  "abstract": "Let $ \\{P_{n}\\}_{n\\geq 0} $ be the sequence of Padovan numbers defined by $ P_0=0 $, $ P_1 = P_2=1$ and $ P_{n+3}= P_{n+1} +P_n$ for all $ n\\geq 0 $. In this paper, we find all positive square-free integers $ d $ such that the Pell equations $ x^2-dy^2 = \\pm 1 $, $ X^2-dY^2=\\pm 4 $ have at least two positive integer solutions $ (x,y) $ and $(x^{\\prime}, y^{\\prime})$, $ (X,Y) $ and $(X^{\\prime}, Y^{\\prime})$, respectively, such that each of $ x, ~x^{\\prime}, ~X, ~X^{\\prime} $ is a sum of two Padovan numbers.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-05-27T16:15:42.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11A25",
      "11B39",
      "11J86"
    ],
    "keywords": [
      "padovan numbers",
      "pell equations",
      "coordinates",
      "positive square-free integers"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 30,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}