{
  "id": "1911.01790",
  "version": "v1",
  "published": "2019-11-01T09:37:04.000Z",
  "updated": "2019-11-01T09:37:04.000Z",
  "title": "Proof of two supercongruences by the Wilf-Zeilberger method",
  "authors": [
    "Guo-Shuai Mao"
  ],
  "comment": "13 pages. arXiv admin note: substantial text overlap with arXiv:1910.09983",
  "categories": [
    "math.NT",
    "math.CO"
  ],
  "abstract": "In this paper, we prove two supercongruences by the Wilf-Zeilberger method. One of them is, for any prime $p>3$, \\begin{align*} \\sum_{n=0}^{(p-1)/2}\\frac{3n+1}{(-8)^n}\\binom{2n}n^3\\equiv p\\left(\\frac{-1}p\\right)+\\frac{p^3}4\\left(\\frac2p\\right)E_{p-3}\\left(\\frac14\\right)\\pmod{p^4}, \\end{align*} where $\\left(\\frac{\\cdot}p\\right)$ stands for the Legendre symbol, and $E_{n}(x)$ are the Euler polynomials. This congruence confirms a conjecture of Sun \\cite[(2.18)]{sun-numb-2019} with $n=1$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-11-01T09:37:04.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "wilf-zeilberger method",
      "supercongruences",
      "congruence confirms",
      "euler polynomials"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 13,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}