Proof of a conjectural supercongruence

Xiang-Zi Meng, Zhi-Wei Sun

Let $m>2$ and $q>0$ be integers with $m$ even or $q$ odd. We show the supercongruence $$\sum _{k=0}^{p-1}(-1{)}^{km}{\left(\genfrac{}{}{0ex}{}{p/m-q}{k}\right)}^m\equiv 0(\mathrm{mod}{p}^3).$$ for any prime $p>mq$. This c…