{
  "id": "1211.4468",
  "version": "v1",
  "published": "2012-11-19T15:45:03.000Z",
  "updated": "2012-11-19T15:45:03.000Z",
  "title": "New Lower Bounds for the Least Common Multiples of Arithmetic Progressions",
  "authors": [
    "Rongjun Wu",
    "Qianrong Tan",
    "Shaofang Hong"
  ],
  "comment": "4 pages. To appear in Chinese Annals of Mathematics (Series B)",
  "journal": "Chin. Ann. Math. 34B (2013), 861-864",
  "categories": [
    "math.NT"
  ],
  "abstract": "For relatively prime positive integers $u_0$ and $r$ and for $0\\le k\\le n$, define $u_k:=u_0+kr$. Let $L_n:={\\rm lcm}(u_0, u_1, ..., u_n)$ and let $a, l\\ge 2$ be any integers. In this paper, we show that, for integers $\\alpha \\geq a$ and $r\\geq \\max(a, l-1)$ and $n\\geq l\\alpha r$, we have $$L_n\\geq u_0r^{(l-1)\\alpha +a-l}(r+1)^n.$$ Particularly, letting $l=3$ yields an improvement to the best previous lower bound on $L_n$ obtained by Hong and Kominers.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2012-11-19T15:45:03.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "lower bound",
      "common multiples",
      "arithmetic progressions",
      "relatively prime positive integers",
      "improvement"
    ],
    "tags": [
      "journal article"
    ],
    "publication": {
      "publisher": "Princeton University and the Institute for Advanced Study",
      "journal": "Ann. Math."
    },
    "note": {
      "typesetting": "TeX",
      "pages": 4,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1211.4468W"
    }
  }
}