Proof of some supercongruences via the Wilf-Zeilberger method

TL;DR

This paper proves certain supercongruences involving binomial coefficients and hypergeometric sums using the Wilf-Zeilberger method, extending known results and establishing new congruence relations for primes and their powers.

Contribution

The paper introduces novel supercongruence proofs for hypergeometric sums using the Wilf-Zeilberger method, expanding the toolkit for congruence analysis in number theory.

Findings

Established supercongruences for sums involving hypergeometric terms.

Extended known supercongruence results to higher prime powers.

Demonstrated the effectiveness of the Wilf-Zeilberger method in proving supercongruences.

Abstract

In this paper, we prove some supercongruences via the Wilf-Zeilberger method. For instance, for any odd prime $p$ and positive integer $r$ and $\delta\in\{1,2\}$, we have \begin{align*} \sum_{n=0}^{(p^r-1)/\delta} \frac{\left(\frac12\right)^5_n}{n!^5}(10n^2+6n+1)(-4)^n &\equiv\begin{cases}p^{2r}\ \pmod{p^{r+4}} &\tt{if}\ r\leq4, \\0\ \pmod{p^{r+4}} &\tt{if}\ r \geq5. \end{cases} \end{align*}