Proof of a supercongruence conjectured by Sun through a $q$-microscope

Abstract

Recently, Z.-W. Sun made the following conjecture: for any odd prime $p$ and odd integer $m$, $$ \frac{1}{m^2{m-1\choose (m-1)/2}}\Bigg(\sum_{k=0}^{(pm-1)/2}\frac{{2k\choose k}}{8^k} -\left(\frac{2}{p}\right)\sum_{k=0}^{(m-1)/2}\frac{{2k\choose k}}{8^k}\Bigg) \equiv 0\pmod{p^2}. $$ In this note, applying the "creative microscoping" method, introduced by the author and Zudilin, we confirm the above conjecture of Sun.