Finding congruences with the WZ method

Abstract

We utilize the Wilf-Zeilberger (WZ) method to establish congruences related to truncated Ramanujan-type series. By constructing hypergeometric terms $f(k,a,b,\dots )$ 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.