{
  "id": "2408.09776",
  "version": "v1",
  "published": "2024-08-19T08:05:38.000Z",
  "updated": "2024-08-19T08:05:38.000Z",
  "title": "Supercongruences via Beukers' method",
  "authors": [
    "Zhi-Hong Sun",
    "Dongxi Ye"
  ],
  "comment": "36 pages",
  "categories": [
    "math.NT",
    "math.CA",
    "math.CO"
  ],
  "abstract": "Recently, using modular forms F. Beukers posed a unified method that can deal with a large number of supercongruences involving binomial coefficients and Ap\\'ery-like numbers. In this paper, we use Beukers' method to prove some conjectures of the first author concerning the congruences for $$\\sum_{k=0}^{(p-1)/2}\\frac{\\binom{2k}k^3}{m^k}, \\quad\\sum_{k=0}^{p-1}\\frac{\\binom{2k}k^2\\binom{4k}{2k}}{m^k}, \\quad \\sum_{k=0}^{p-1}\\frac{\\binom{2k}k\\binom{3k}k\\binom{6k}{3k}}{m^k}, \\quad \\sum_{k=0}^{p-1}\\frac{V_k}{m^k} \\quad\\hbox{and}\\quad \\sum_{k=0}^{p-1}\\frac{T_k}{m^k} $$ modulo $p^3$, where $p>3$ is a prime, $m$ is an integer not divisible by $p$, $V_n=\\sum_{k=0}^n\\binom{2k}k^2\\binom{2n-2k}{n-k}^2$ and $T_n=\\sum_{k=0}^n\\binom nk^2\\binom{2k}n^2$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-08-19T08:05:38.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11A07",
      "11B65",
      "11E25",
      "11F03",
      "11F20"
    ],
    "keywords": [
      "supercongruences",
      "modular forms",
      "binomial coefficients",
      "large number",
      "conjectures"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 36,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}