{
  "id": "2506.19221",
  "version": "v1",
  "published": "2025-06-24T01:03:11.000Z",
  "updated": "2025-06-24T01:03:11.000Z",
  "title": "Finding congruences with the WZ method",
  "authors": [
    "Li-Quan Feng",
    "Qing-Hu Hou"
  ],
  "categories": [
    "math.CO"
  ],
  "abstract": "We utilize the Wilf-Zeilberger (WZ) method to establish congruences related to truncated Ramanujan-type series. By constructing hypergeometric terms $f(k, a, b, \\ldots)$ with Gosper-summable differences and selecting appropriate parameters, we derive several congruences modulo $p$ and $p^2$ for primes $p > 2$. For instance, we prove that for any prime $p > 2$, \\[ \\sum_{n=0}^{p-1} \\frac{10n+3}{2^{3n}}\\binom{3n}{n}\\binom{2n}{n}^2 \\equiv 0 \\pmod{p},\\] and \\[ \\sum_{n=0}^{p-1} \\frac{(-1)^n(20n^2+8n+1)}{2^{12n}}\\binom{2n}{n}^5 \\equiv 0 \\pmod{p^2}. \\] These results partially confirm conjectures by Sun and provide some novel congruences.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2025-06-24T01:03:11.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "wz method",
      "finding congruences",
      "results partially confirm conjectures",
      "novel congruences",
      "truncated ramanujan-type series"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}