{
  "id": "2111.08775",
  "version": "v3",
  "published": "2021-11-16T20:52:30.000Z",
  "updated": "2021-12-03T14:05:14.000Z",
  "title": "On two congruence conjectures of Z.-W. Sun involving Franel numbers",
  "authors": [
    "Guo-Shuai Mao",
    "Yan Liu"
  ],
  "comment": "20 pages, revised some typos",
  "journal": "Proceedings of the Royal Society of Edinburgh Section A-Mathematics (2024)",
  "doi": "10.1017/prm.2023.41",
  "categories": [
    "math.NT",
    "math.CO"
  ],
  "abstract": "In this paper, we mainly prove the following conjectures of Z.-W. Sun \\cite{S13}: Let $p>2$ be a prime. If $p=x^2+3y^2$ with $x,y\\in\\mathbb{Z}$ and $x\\equiv1\\pmod 3$, then $$x\\equiv\\frac14\\sum_{k=0}^{p-1}(3k+4)\\frac{f_k} {2^k}\\equiv\\frac12\\sum_{k=0}^{p-1}(3k+2)\\frac{f_k}{(-4)^k}\\pmod{p^2},$$ and if $p\\equiv1\\pmod3$, then $$\\sum_{k=0}^{p-1}\\frac{f_k}{2^k}\\equiv\\sum_{k=0}^{p-1}\\frac{f_k}{(-4)^k}\\pmod{p^3},$$ where $f_n=\\sum_{k=0}^n\\binom{n}k^3$ stands for the $n$th Franel number.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2021-12-03T14:05:14.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "congruence conjectures",
      "th franel number"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 20,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}