{
  "id": "1906.07652",
  "version": "v1",
  "published": "2019-06-16T07:51:23.000Z",
  "updated": "2019-06-16T07:51:23.000Z",
  "title": "On the divisibility of binomial coefficients",
  "authors": [
    "Sílvia Casacuberta"
  ],
  "categories": [
    "math.NT"
  ],
  "abstract": "In Pacific J. Math. 292 (2018), 223-238, Shareshian and Woodroofe asked if for every positive integer $n$ there exist primes $p$ and $q$ such that, for all integers $k$ with $1 \\leq k \\leq n-1$, the binomial coefficient $\\binom{n}{k}$ is divisible by at least one of $p$ or $q$. We give conditions under which a number $n$ has this property and discuss a variant of this problem involving more than two primes. We prove that every positive integer $n$ has infinitely many multiples with this property.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-06-16T07:51:23.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11B65"
    ],
    "keywords": [
      "binomial coefficient",
      "divisibility",
      "positive integer",
      "shareshian"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}