{
  "id": "2103.05830",
  "version": "v1",
  "published": "2021-03-10T02:24:04.000Z",
  "updated": "2021-03-10T02:24:04.000Z",
  "title": "Three supercongruences for Apery numbers or Franel numbers",
  "authors": [
    "Yong Zhang"
  ],
  "categories": [
    "math.NT",
    "math.CO"
  ],
  "abstract": "The Ap\\'ery numbers $A_n$ and the Franel numbers $f_n$ are defined by $$A_n=\\sum_{k=0}^{n}{\\binom{n+k}{2k}}^2{\\binom{2k}{k}}^2\\ \\ \\ \\ \\ {\\rm and }\\ \\ \\ \\ \\ \\ f_n=\\sum_{k=0}^{n}{\\binom{n}{k}}^3(n=0, 1, \\cdots,).$$ In this paper, we prove three supercongruences for Ap\\'ery numbers or Franel numbers conjectured by Z.-W. Sun. Let $p\\geq 5$ be a prime and let $n\\in \\mathbb{Z}^{+}$. We show that \\begin{align} \\notag \\frac{1}{n}\\bigg(\\sum_{k=0}^{pn-1}(2k+1)A_k-p\\sum_{k=0}^{n-1}(2k+1)A_k\\bigg)\\equiv0\\pmod{p^{4+3\\nu_p(n)}} \\end{align} and \\begin{align}\\notag \\frac{1}{n^3}\\bigg(\\sum_{k=0}^{pn-1}(2k+1)^3A_k-p^3\\sum_{k=0}^{n-1}(2k+1)^3A_k\\bigg)\\equiv0\\pmod{p^{6+3\\nu_p(n)}}, \\end{align} where $\\nu_p(n)$ denotes the $p$-adic order of $n$. Also, for any prime $p$ we have \\begin{align} \\notag \\frac{1}{n^3}\\bigg(\\sum_{k=0}^{pn-1}(3k+2)(-1)^kf_k-p^2\\sum_{k=0}^{n-1}(3k+2)(-1)^kf_k\\bigg)\\equiv0\\pmod{p^{3}}. \\end{align}",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-03-10T02:24:04.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11A07",
      "11B65",
      "05A10",
      "11B39",
      "11B75"
    ],
    "keywords": [
      "apery numbers",
      "franel numbers",
      "supercongruences"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}