{
  "id": "0803.0290",
  "version": "v1",
  "published": "2008-03-03T17:21:56.000Z",
  "updated": "2008-03-03T17:21:56.000Z",
  "title": "Nontrivial lower bounds for the least common multiple of some finite sequences of integers",
  "authors": [
    "Bakir Farhi"
  ],
  "comment": "16 pages, published in Journal of Number Theory",
  "journal": "J. Number Theory, 125 (2007), p. 393-411",
  "categories": [
    "math.NT"
  ],
  "abstract": "We present here a method which allows to derive a nontrivial lower bounds for the least common multiple of some finite sequences of integers. We obtain efficient lower bounds (which in a way are optimal) for the arithmetic progressions and lower bounds less efficient (but nontrivial) for quadratic sequences whose general term has the form $u_n = a n (n + t) + b$ with $(a, t, b) \\in {\\mathbb{Z}}^3, a \\geq 5, t \\geq 0, \\rm{gcd}(a, b) = 1$. From this, we deduce for instance the lower bound: $\\mathrm{lcm}\\{1^2 + 1, 2^2 + 1, ..., n^2 + 1\\} \\geq 0,32 (1,442)^n$ (for all $n \\geq 1$). In the last part of this article, we study the integer $\\mathrm{lcm}(n, n + 1, ..., n + k)$ $(k \\in \\mathbb{N}, n \\in {\\mathbb{N}}^*)$. We show that it has a divisor $d_{n, k}$ simple in its dependence on $n$ and $k$, and a multiple $m_{n, k}$ also simple in its dependence on $n$. In addition, we prove that both equalities: $\\mathrm{lcm}(n, n + 1, ..., n + k) = d_{n, k}$ and $\\mathrm{lcm}(n, n + 1, ..., n + k) = m_{n, k}$ hold for an infinitely many pairs $(n, k)$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2008-03-03T17:21:56.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11A05",
      "11B25"
    ],
    "keywords": [
      "nontrivial lower bounds",
      "finite sequences",
      "common multiple",
      "efficient lower bounds",
      "dependence"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 16,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2008arXiv0803.0290F"
    }
  }
}