{
  "id": "0808.1507",
  "version": "v1",
  "published": "2008-08-11T12:48:56.000Z",
  "updated": "2008-08-11T12:48:56.000Z",
  "title": "New results on the least common multiple of consecutive integers",
  "authors": [
    "Bakir Farhi",
    "Daniel Kane"
  ],
  "comment": "8 pages, to appear",
  "categories": [
    "math.NT"
  ],
  "abstract": "When studying the least common multiple of some finite sequences of integers, the first author introduced the interesting arithmetic functions $g_k$ $(k \\in \\mathbb{N})$, defined by $g_k(n) := \\frac{n (n + 1) ... (n + k)}{\\lcm(n, n + 1, >..., n + k)}$ $(\\forall n \\in \\mathbb{N} \\setminus \\{0\\})$. He proved that $g_k$ $(k \\in \\mathbb{N})$ is periodic and $k!$ is a period of $g_k$. He raised the open problem consisting to determine the smallest positive period $P_k$ of $g_k$. Very recently, S. Hong and Y. Yang have improved the period $k!$ of $g_k$ to $\\lcm(1, 2, ..., k)$. In addition, they have conjectured that $P_k$ is always a multiple of the positive integer $\\frac{\\lcm(1, 2, >..., k, k + 1)}{k + 1}$. An immediate consequence of this conjecture states that if $(k + 1)$ is prime then the exact period of $g_k$ is precisely equal to $\\lcm(1, 2, ..., k)$. In this paper, we first prove the conjecture of S. Hong and Y. Yang and then we give the exact value of $P_k$ $(k \\in \\mathbb{N})$. We deduce, as a corollary, that $P_k$ is equal to the part of $\\lcm(1, 2, ..., k)$ not divisible by some prime.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2008-08-11T12:48:56.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11A05"
    ],
    "keywords": [
      "common multiple",
      "consecutive integers",
      "open problem",
      "finite sequences",
      "smallest positive period"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 8,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}