On a supercongruence conjecture of Z.-W. Sun

Guo-Shuai Mao

Published 2020-03-30Version 1

In this paper, we prove a supercongruence conjectured by Z.-W. Sun in 2013. The conjecture states that: Let $p$ be an odd prime and let $a\in\mathbb{Z}^{+}$. Then if $p\equiv1\pmod3$ or $a>1$, we have \begin{align*} \sum_{k=0}^{\lfloor\frac{5}6p^a\rfloor}\frac{\binom{2k}k}{16^k}\equiv\left(\frac{3}{p^a}\right)\pmod{p^2}, \end{align*} where $\left(\frac{\cdot}{\cdot}\right)$ is the Jacobi symbol.