{
  "id": "1609.00440",
  "version": "v1",
  "published": "2016-09-02T01:00:39.000Z",
  "updated": "2016-09-02T01:00:39.000Z",
  "title": "On the subgroup generated by solutions of Pell's equation",
  "authors": [
    "Elena C. Covill",
    "Mohammad Javaheri",
    "Nikolai A. Krylov"
  ],
  "comment": "12 pages",
  "categories": [
    "math.NT"
  ],
  "abstract": "Equivalence classes of solutions of the Diophantine equation $a^2+mb^2=c^2$ form an infinitely generated abelian group $G_m$ under the operation induced by complex multiplication, where $m$ is a fixed square-free positive integer. Solutions of Pell's equation $x^2-my^2=1$ generate a subgroup $P_m$ of $G_m$. We prove that $G_m/P_m$ has infinite rank for infinitely many values of $m$. We also give several examples of $m$ for which $G_m/P_m$ has nontrivial torsion.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-09-02T01:00:39.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11D09",
      "11R11",
      "11A55"
    ],
    "keywords": [
      "pells equation",
      "diophantine equation",
      "infinitely generated abelian group",
      "nontrivial torsion",
      "equivalence classes"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 12,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}