Proof of a conjectural supercongruence modulo $p^5$

Abstract

In this paper we prove the supercongruence $$\sum _{n=0}^{(p-1)/2}\frac{6n+1}{256^n}{\left(\genfrac{}{}{0ex}{}{2n}{n}\right)}^3\equiv p(-1{)}^{(p-1)/2}+(-1{)}^{(p-1)/2}\frac{7}{24}{p}^4{B}_{p-3}(\mathrm{mod}{p}^5)$$ for any prime $p>3$, which was conjectured by Sun in 2019.