Extension of a conjectural supercongruence of (G.3) of Swisher using Zeilberger's algorithm

TL;DR

This paper proves a supercongruence involving binomial coefficients and hypergeometric terms using Zeilberger's algorithm, extending Swisher's conjecture to higher prime powers for odd integers greater than 3.

Contribution

It provides a proof of a supercongruence that extends Swisher's conjecture to higher prime powers using Zeilberger's algorithm.

Findings

Confirmed supercongruence holds for higher prime powers.

Extended the conjecture to cases with r > 3.

Demonstrated the effectiveness of Zeilberger's algorithm in proving supercongruences.

Abstract

Using Zeilberger's algorithm, we here give a proof of the supercongruence $$ \sum_{n=0}^{\frac{p^r-3}{4}}(8n+1)\frac{\left(\frac{1}{4}\right)_n^4}{(1)_n^4}\equiv -p^3 \sum_{n=0}^{\frac{p^{r-2}-3}{4}}(8n+1)\frac{\left(\frac{1}{4}\right)_n^4}{(1)_n^4} ~~(\text{mod }p^{\frac{3r-1}{2}}),$$ for any odd integer $r>3$. This extends the third conjectural supercongruence of (G.3) of Swisher to modulo higher prime powers than that expected by Swisher.