{
  "id": "1011.3487",
  "version": "v8",
  "published": "2010-11-15T19:59:35.000Z",
  "updated": "2015-02-26T15:49:34.000Z",
  "title": "Supercongruences motivated by e",
  "authors": [
    "Zhi-Wei Sun"
  ],
  "comment": "16 pages",
  "journal": "J. Number Theory 147(2015), 326-341",
  "categories": [
    "math.NT",
    "math.CO"
  ],
  "abstract": "In this paper we establish some new supercongruences motivated by the well-known fact $\\lim_{n\\to\\infty}(1+1/n)^n=e$. Let $p>3$ be a prime. We prove that $$\\sum_{k=0}^{p-1}\\binom{-1/(p+1)}k^{p+1}\\equiv 0\\ \\pmod{p^5}\\ \\ \\ \\mbox{and}\\ \\ \\ \\sum_{k=0}^{p-1}\\binom{1/(p-1)}k^{p-1}\\equiv \\frac{2}{3}p^4B_{p-3}\\ \\pmod{p^5},$$ where $B_0,B_1,B_2,\\ldots$ are Bernoulli numbers. We also show that for any $a\\in\\mathbb Z$ with $p\\nmid a$ we have $$\\sum_{k=1}^{p-1}\\frac1k\\left(1+\\frac ak\\right)^k\\equiv -1\\pmod{p}\\ \\ \\ \\mbox{and}\\ \\ \\ \\sum_{k=1}^{p-1}\\frac1{k^2}\\left(1+\\frac ak\\right)^k\\equiv 1+\\frac 1{2a}\\pmod{p}.$$",
  "revisions": [
    {
      "version": "v7",
      "updated": "2014-07-21T14:19:56.000Z",
      "abstract": "In this paper we establish some new supercongruences motivated by the well-known fact $\\lim_{n\\to\\infty}(1+1/n)^n=e$. Let $p>3$ be a prime. We show that $$\\sum_{k=0}^{p-1}\\binom{-1/(p+1)}k^{p+1}\\equiv 0\\ \\pmod{p^5}\\ \\ \\ \\mbox{and}\\ \\ \\ \\sum_{k=0}^{p-1}\\binom{1/(p-1)}k^{p-1}\\equiv \\frac{2}{3}p^4B_{p-3}\\ \\pmod{p^5},$$ where $B_0,B_1,B_2,\\ldots$ are Bernoulli numbers. For any integer $m\\not\\equiv0\\pmod{p}$, we prove that $$\\sum_{k=0}^{p-1}(-1)^{km}\\binom{p/m-1}k^m\\equiv\\frac{(m-1)(7m-5)}{36m^2}p^4B_{p-3}\\pmod{p^5},$$ and $$\\sum_{k=1}^{p-1}\\frac{(-1)^{km}}{k^2}\\binom{p/m-1}k^m\\equiv\\frac 1p\\sum_{k=1}^{p-1}\\frac 1k\\ \\pmod{p^3}\\ \\ \\ \\ \\mbox{if}\\ p>5.$$ We also show that for any $a\\in\\mathbb Z$ with $p\\nmid a$ we have $$\\sum_{k=1}^{p-1}\\frac1k\\left(1+\\frac ak\\right)^k\\equiv -1\\pmod{p}\\ \\ \\ \\mbox{and}\\ \\ \\ \\sum_{k=1}^{p-1}\\frac1{k^2}\\left(1+\\frac ak\\right)^k\\equiv 1+\\frac 1{2a}\\pmod{p}.$$",
      "comment": "16 pages, revised version for publication. See Theorems 1.1-1.2 for new additions",
      "journal": null,
      "doi": null
    },
    {
      "version": "v8",
      "updated": "2015-02-26T15:49:34.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11B65",
      "11A07",
      "05A10",
      "11B68"
    ],
    "keywords": [
      "supercongruences",
      "well-known fact",
      "bernoulli numbers"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 16,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2010arXiv1011.3487S"
    }
  }
}