In this paper, we confirm several conjectures posed by Sun recently; for example, we prove that for any odd prime $p$ we have $$ \sum_{k=0}^{p-1}A_k\equiv\begin{cases}4x^2-2p\pmod{p^2}\quad&\text{if $p=x^2+2y^2\ (x,y\in\mathbb{Z})$},\\ 0\pmod{p^2}\quad&\text{if $p\equiv5,7\pmod{8}$},\end{cases} $$ where $A_n:=\sum_{k=0}^n\binom{n+k}{k}^2\binom{n}{k}^2$ are the Apéry numbers.
此内容由惯性聚合(RSS阅读器)自动聚合整理，仅供阅读参考。 原文来自 — 版权归原作者所有。