{
  "id": "1706.09804",
  "version": "v1",
  "published": "2017-06-29T15:27:56.000Z",
  "updated": "2017-06-29T15:27:56.000Z",
  "title": "Denominators of Bernoulli polynomials",
  "authors": [
    "Olivier Bordellès",
    "Florian Luca",
    "Pieter Moree",
    "Igor E. Shparlinski"
  ],
  "comment": "25 pages",
  "categories": [
    "math.NT"
  ],
  "abstract": "For a positive integer $n$ let $\\mathfrak{P}_n=\\prod_{s_p(n)\\ge p} p,$ where $p$ runs over all primes and $s_p(n)$ is the sum of the base $p$ digits of $n$. For all $n$ we prove that $\\mathfrak{P}_n$ is divisible by all \"small\" primes with at most one exception. We also show that $\\mathfrak{P}_n$ is large, has many prime factors exceeding $\\sqrt{n}$, with the largest one exceeding $n^{20/37}$. We establish Kellner's conjecture, which says that the number of prime factors exceeding $\\sqrt{n}$ grows asymptotically as $\\kappa \\sqrt{n}/\\log n$ for some constant $\\kappa$ with $\\kappa=2$. Further, we compare the sizes of $\\mathfrak{P}_n$ and $\\mathfrak{P}_{n+1}$, leading to the somewhat surprising conclusion that although $\\mathfrak{P}_n$ tends to infinity with $n$, the inequality $\\mathfrak{P}_n>\\mathfrak{P}_{n+1}$ is more frequent than its reverse.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-06-29T15:27:56.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11N37",
      "11J68",
      "11B68"
    ],
    "keywords": [
      "bernoulli polynomials",
      "denominators",
      "prime factors exceeding",
      "somewhat surprising conclusion",
      "establish kellners conjecture"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 25,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}