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

Abstract: Using Zeilberger's algorithm, we here give a proof of the supercongruence $$\sum_{n=0}^{{\frac{p^{r-3}{4}}(8n+1)\frac{\left(\frac{1}{4}\right)_n^{4}{(1)_n^4}\equiv}}} -p³ \sum_{n=0}^{{\frac{p^{{r-2}-3}{4}}(8n+1)\frac{\left(\frac{1}{4}\right)_n^{4}{(1)_n^4}}}} ~~(\text{mod }p<sup>{\frac{3r-1}{2}}),$$ for any odd integer $r&gt;3$. This extends the third conjectural supercongruence of (G.3) of Swisher to modulo higher prime powers than that expected by Swisher.