{
  "id": "0903.0530",
  "version": "v3",
  "published": "2009-03-03T13:31:38.000Z",
  "updated": "2010-02-18T17:09:25.000Z",
  "title": "The least common multiple of consecutive arithmetic progression terms",
  "authors": [
    "Shaofang Hong",
    "Guoyou Qian"
  ],
  "comment": "10 pages. To appear in Proceedings of the Edinburgh Mathematical Society",
  "journal": "Proceedings of the Edinburgh Mathematical Society 54 (2011), 431-441",
  "doi": "10.1017/S0013091509000431",
  "categories": [
    "math.NT"
  ],
  "abstract": "Let $k\\ge 0,a\\ge 1$ and $b\\ge 0$ be integers. We define the arithmetic function $g_{k,a,b}$ for any positive integer $n$ by $g_{k,a,b}(n):=\\frac{(b+na)(b+(n+1)a)...(b+(n+k)a)} {{\\rm lcm}(b+na,b+(n+1)a,...,b+(n+k)a)}.$ Letting $a=1$ and $b=0$, then $g_{k,a,b}$ becomes the arithmetic function introduced previously by Farhi. Farhi proved that $g_{k,1,0}$ is periodic and that $k!$ is a period. Hong and Yang improved Farhi's period $k!$ to ${\\rm lcm}(1,2,...,k)$ and conjectured that $\\frac{{\\rm lcm}(1,2,...,k,k+1)}{k+1}$ divides the smallest period of $g_{k,1,0}$. Recently, Farhi and Kane proved this conjecture and determined the smallest period of $g_{k,1,0}$. For the general integers $a\\ge 1$ and $b\\ge 0$, it is natural to ask the interesting question: Is $g_{k,a,b}$ periodic? If so, then what is the smallest period of $g_{k,a,b}$? We first show that the arithmetic function $g_{k,a,b}$ is periodic. Subsequently, we provide detailed $p$-adic analysis of the periodic function $g_{k,a,b}$. Finally, we determine the smallest period of $g_{k,a,b}$. Our result extends the Farhi-Kane theorem from the set of positive integers to general arithmetic progressions.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2010-02-18T17:09:25.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11B25",
      "11N13",
      "11A05"
    ],
    "keywords": [
      "consecutive arithmetic progression terms",
      "smallest period",
      "common multiple",
      "arithmetic function",
      "positive integer"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 10,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2009arXiv0903.0530H"
    }
  }
}