{
  "id": "1301.5944",
  "version": "v2",
  "published": "2013-01-25T01:53:37.000Z",
  "updated": "2015-07-08T14:22:40.000Z",
  "title": "On a theorem of Serret on continued fractions",
  "authors": [
    "Paloma Bengoechea"
  ],
  "categories": [
    "math.NT"
  ],
  "abstract": "A classical theorem in continued fractions due to Serret shows that for any two irrational numbers x and y related by a transformation $\\gamma$ in PGL(2,Z) there exist s and t for which the complete quotients x_s and y_t coincide. In this paper we give an upper bound in terms of $\\gamma$ for the smallest indices s and t.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2013-01-25T01:53:37.000Z",
      "title": "A refinement of a classic theorem on continued fractions",
      "abstract": "We express the set of transformations occuring in two different continued fraction algorithms as subspaces of PGL(2,Z) defined by certain simple linear inequalities. As a consequence, we improve a Hurwitz's classic theorem on continued fractions giving, for $\\gamma$ in PGL(2,Z), a bound depending only on $\\gamma$ for the index of the term from which the continued fractions of two irrational numbers related by $\\gamma$ start being identical.",
      "comment": "10 pages",
      "journal": null,
      "doi": null,
      "authors": [
        "P. Bengoechea"
      ]
    },
    {
      "version": "v2",
      "updated": "2015-07-08T14:22:40.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "refinement",
      "hurwitzs classic theorem",
      "simple linear inequalities",
      "continued fraction algorithms",
      "irrational numbers"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 10,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1301.5944B"
    }
  }
}