{
  "id": "2008.10294",
  "version": "v1",
  "published": "2020-08-24T09:54:01.000Z",
  "updated": "2020-08-24T09:54:01.000Z",
  "title": "Nontrivial effective lower bounds for the least common multiple of a $q$-arithmetic progression",
  "authors": [
    "Bakir Farhi"
  ],
  "comment": "15 pages",
  "categories": [
    "math.NT"
  ],
  "abstract": "This paper is devoted to establish nontrivial effective lower bounds for the least common multiple of consecutive terms of a sequence ${(u_n)}_{n \\in \\mathbb{N}}$ whose general term has the form $u_n = r {[n]}_q + u_0$, where $q , r$ are positive integers and $u_0$ is a non-negative integer such that $\\mathrm{gcd}(u_0 , r) = \\mathrm{gcd}(u_1 , q) = 1$. For such a sequence, we show that for all positive integer $n$, we have $\\mathrm{lcm}\\{u_1 , u_2 , \\dots , u_n\\} \\geq c_1 \\cdot c_2^n \\cdot q^{\\frac{n^2}{4}}$, where $c_1$ and $c_2$ are positive constants depending only on $q , r$ and $u_0$. This can be considered as a $q$-analog of the lower bounds already obtained by the author (in 2005) and by Hong and Feng (in 2006) for the arithmetic progressions.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-08-24T09:54:01.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11A05",
      "11B25",
      "11B65",
      "05A30"
    ],
    "keywords": [
      "arithmetic progression",
      "common multiple",
      "positive integer",
      "establish nontrivial effective lower bounds",
      "general term"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 15,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}