{
  "id": "1910.13218",
  "version": "v1",
  "published": "2019-10-29T12:01:42.000Z",
  "updated": "2019-10-29T12:01:42.000Z",
  "title": "A lower bound on the LCM of polynomial sequences",
  "authors": [
    "James Maynard",
    "Zeev Rudnick"
  ],
  "comment": "7 pages",
  "categories": [
    "math.NT"
  ],
  "abstract": "Let $f$ be a polynomial $f$ of degree $d\\ge 2$ with integer coefficients which is irreducible over the rationals. Cilleruelo conjectured that the least common multiple of the values of the polynomial at the first $N$ integers satisfies $\\log lcm(f(1),\\dots, f(N)) \\sim (d-1) N\\log N$ as $N\\to \\infty$. This is only known for degree $d=2$. In this note we give a simple lower bound for all degrees $d\\geq 2$ which is consistent with the conjecture: $\\log lcm (f(1),\\dots, f(N)) \\gg N\\log N$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-10-29T12:01:42.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11A05"
    ],
    "keywords": [
      "polynomial sequences",
      "simple lower bound",
      "integer coefficients",
      "common multiple",
      "integers satisfies"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 7,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}