  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 Ap\'{e}ry numbers.
