{
  "id": "2103.07967",
  "version": "v1",
  "published": "2021-03-14T16:32:50.000Z",
  "updated": "2021-03-14T16:32:50.000Z",
  "title": "On the least common multiple of shifted powers",
  "authors": [
    "Carlo Sanna"
  ],
  "categories": [
    "math.NT",
    "math.PR"
  ],
  "abstract": "Let $a \\geq 2$ be an integer. We prove that for every periodic sequence $(s_n)_{n \\geq 1}$ in $\\{-1, +1\\}$ there exists an effectively computable rational number $C_\\mathbf{s} > 0$ such that \\begin{equation*} \\log\\operatorname{lcm}(a + s_1, a^2 + s_2, \\dots, a^n + s_n) \\sim C_\\mathbf{s} \\cdot \\frac{\\log a}{\\pi^2} \\cdot n^2 , \\end{equation*} as $n \\to +\\infty$, where $\\operatorname{lcm}$ denotes the least common multiple. Furthermore, we show that if $(s_n)_{n \\geq 1}$ is a sequence of independent and uniformly distributed random variables in $\\{-1, +1\\}$ then \\begin{equation*} \\log\\operatorname{lcm}(a + s_1, a^2 + s_2, \\dots, a^n + s_n) \\sim 6 \\operatorname{Li}_2\\!\\big(\\tfrac1{2}\\big) \\cdot \\frac{\\log a}{\\pi^2} \\cdot n^2 , \\end{equation*} with probability $1 - o(1)$, as $n \\to +\\infty$, where $\\operatorname{Li}_2$ is the dilogarithm function.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-03-14T16:32:50.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11B39",
      "11B37",
      "11N37"
    ],
    "keywords": [
      "common multiple",
      "shifted powers",
      "dilogarithm function",
      "periodic sequence",
      "effectively computable rational number"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}