English

Finding congruences with the WZ method

Combinatorics 2025-06-25 v1

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,)f(k, a, b, \ldots) with Gosper-summable differences and selecting appropriate parameters, we derive several congruences modulo pp and p2p^2 for primes p>2p > 2. For instance, we prove that for any prime p>2p > 2, n=0p110n+323n(3nn)(2nn)20(modp), \sum_{n=0}^{p-1} \frac{10n+3}{2^{3n}}\binom{3n}{n}\binom{2n}{n}^2 \equiv 0 \pmod{p}, and n=0p1(1)n(20n2+8n+1)212n(2nn)50(modp2). \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.

Keywords

Cite

@article{arxiv.2506.19221,
  title  = {Finding congruences with the WZ method},
  author = {Li-Quan Feng and Qing-Hu Hou},
  journal= {arXiv preprint arXiv:2506.19221},
  year   = {2025}
}
R2 v1 2026-07-01T03:30:37.609Z