{
  "id": "1303.2887",
  "version": "v1",
  "published": "2013-03-12T14:15:26.000Z",
  "updated": "2013-03-12T14:15:26.000Z",
  "title": "The period length of Euler's number e",
  "authors": [
    "Kurt Girstmair"
  ],
  "comment": "11 pages",
  "categories": [
    "math.NT"
  ],
  "abstract": "Let s_k/t_k, k>= 0, be the convergents of the continued fraction expansion of a real number x. We investigate the sequence of Jacobi symbols (s_k/t_k), k>= 0. We show that this sequence is purely periodic with shortest possible period length 24 for x=e=2.718281... and shortest possible period length 40 for x=e^2. Further, we make the first steps towards a general theory of such sequences of Jacobi symbols. For instance, we show that there are uncountably many numbers x such that this sequence has the period 1 (of length 1), and that every natural number L actually occurs as the shortest possible period length of some x.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2013-03-12T14:15:26.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11A55",
      "11A15"
    ],
    "keywords": [
      "period length",
      "eulers number",
      "jacobi symbols",
      "first steps",
      "real number"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 11,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1303.2887G"
    }
  }
}