{
  "id": "2007.13330",
  "version": "v1",
  "published": "2020-07-27T07:09:43.000Z",
  "updated": "2020-07-27T07:09:43.000Z",
  "title": "On the l.c.m. of shifted Fibonacci numbers",
  "authors": [
    "Carlo Sanna"
  ],
  "categories": [
    "math.NT"
  ],
  "abstract": "Let $(F_n)_{n \\geq 1}$ be the sequence of Fibonacci numbers. Guy and Matiyasevich proved that \\begin{equation*} \\log \\operatorname{lcm} (F_1, F_2, \\dots, F_n) \\sim \\frac{3 \\log \\alpha}{\\pi^2} \\cdot n^2 \\quad \\text{as } n \\to +\\infty, \\end{equation*} where $\\operatorname{lcm}$ is the least common multiple and $\\alpha := \\big(1 + \\sqrt{5}) / 2$ is the golden ratio. We prove that for every periodic sequence $\\mathbf{s} = (s_n)_{n \\geq 1}$ in $\\{-1,+1\\}$ there exists an effectively computable rational number $C_{\\mathbf{s}} > 0$ such that \\begin{equation*} \\log \\operatorname{lcm} (F_3 + s_3, F_4 + s_4, \\dots, F_n + s_n) \\sim \\frac{3 \\log \\alpha}{\\pi^2} \\cdot C_\\mathbf{s} \\cdot n^2 , \\quad \\text{as } n \\to +\\infty . \\end{equation*} Moreover, we show that if $(s_n)_{n \\geq 1}$ is a sequence of independent uniformly distributed random variables in $\\{-1,+1\\}$ then \\begin{equation*} \\mathbb{E}\\big[\\log \\operatorname{lcm} (F_3 + s_3, F_4 + s_4, \\dots, F_n + s_n)\\big] \\sim \\frac{3 \\log \\alpha}{\\pi^2} \\cdot \\frac{15 \\operatorname{Li}_2(1 / 16)}{2} \\cdot n^2 , \\quad \\text{as } n \\to +\\infty , \\end{equation*} where $\\operatorname{Li}_2$ is the dilogarithm function.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-07-27T07:09:43.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11B39",
      "11B37",
      "11N37"
    ],
    "keywords": [
      "shifted fibonacci numbers",
      "independent uniformly distributed random variables",
      "periodic sequence",
      "effectively computable rational number",
      "common multiple"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}