{
  "id": "1909.04342",
  "version": "v1",
  "published": "2019-09-10T08:02:42.000Z",
  "updated": "2019-09-10T08:02:42.000Z",
  "title": "On the l.c.m. of random terms of binary recurrence sequences",
  "authors": [
    "Carlo Sanna"
  ],
  "categories": [
    "math.NT"
  ],
  "abstract": "For every positive integer $n$ and every $\\delta \\in [0,1]$, let $B(n, \\delta)$ denote the probabilistic model in which a random set $A \\subseteq \\{1, \\dots, n\\}$ is constructed by choosing independently every element of $\\{1, \\dots, n\\}$ with probability $\\delta$. Moreover, let $(u_k)_{k \\geq 0}$ be an integer sequence satisfying $u_k = a_1 u_{k - 1} + a_2 u_{k - 2}$, for every integer $k \\geq 2$, where $u_0 = 0$, $u_1 \\neq 0$, and $a_1, a_2$ are fixed nonzero integers; and let $\\alpha$ and $\\beta$, with $|\\alpha| \\geq |\\beta|$, be the two roots of the polynomial $X^2 - a_1 X - a_2$. Also, assume that $\\alpha / \\beta$ is not a root of unity. We prove that, as $\\delta n / \\log n \\to +\\infty$, for every $A$ in $B(n, \\delta)$ we have $$\\log \\operatorname{lcm} (u_a : a \\in A) \\sim \\frac{\\delta\\operatorname{Li}_2(1 - \\delta)}{1 - \\delta} \\cdot \\frac{3\\log\\!\\big|\\alpha / \\!\\sqrt{(a_1^2, a_2)}\\big|}{\\pi^2} \\cdot n^2 $$ with probability $1 - o(1)$, where $\\operatorname{lcm}$ denotes the lowest common multiple, $\\operatorname{Li}_2$ is the dilogarithm, and the factor involving $\\delta$ is meant to be equal to $1$ when $\\delta = 1$. This extends previous results of Akiyama, Tropak, Matiyasevich, Guy, Kiss and M\\'aty\\'as, who studied the deterministic case $\\delta = 1$, and is motivated by an asymptotic formula for $\\operatorname{lcm}(A)$ due to Cilleruelo, Ru\\'{e}, \\v{S}arka, and Zumalac\\'{a}rregui.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-09-10T08:02:42.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11B37",
      "11N37"
    ],
    "keywords": [
      "binary recurrence sequences",
      "random terms",
      "lowest common multiple",
      "deterministic case",
      "probabilistic model"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}