On a supercongruence conjecture of Z.-W. Sun

Abstract

In this paper, we partly prove a supercongruence conjectured by Z.-W. Sun in 2013. Let $p$ be an odd prime and let $a\in {\mathbb{Z}}^{+}$. Then if $p\equiv 1(\mathrm{mod}3)$, 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.

8 pages

References in corpus (2)

• Congruences on sums of $q$-binomial coefficients
• Proof of a conjecture of Adamchuk

Cited by in corpus (1)

• Three pairs of congruences concerning sums of central binomial coefficients