{
  "id": "2111.08778",
  "version": "v2",
  "published": "2021-11-16T20:58:29.000Z",
  "updated": "2023-05-31T22:36:09.000Z",
  "title": "Proof of some congruence conjectures of Z.-H. Sun involving Apéry-like numbers",
  "authors": [
    "Guo-Shuai Mao"
  ],
  "comment": "12 pages",
  "categories": [
    "math.NT",
    "math.CO"
  ],
  "abstract": "In this paper, we mainly prove the following conjecture of Z.-H. Sun cite{SH20}: Let $p>3$ be a prime. Then $$\\sum_{k=0}^{p-1}\\binom{2k}k\\frac{3k+1}{(-16)^k}f_k\\equiv(-1)^{(p-1)/2}p+p^3E_{p-3}\\pmod{p^4},$$ where $f_n=\\sum_{k=0}^n\\binom{n}k^3$ and $E_n$ stand for the $n$th Franel number and $n$th Euler number respectively.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2023-05-31T22:36:09.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "congruence conjectures",
      "apéry-like numbers",
      "th euler number",
      "th franel number",
      "sun cite"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 12,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}