Proof of some congruence conjectures of Z.-H. Sun involving Apéry-like numbers

Guo-Shuai Mao

Published 2021-11-16, updated 2023-05-31Version 2

In this paper, we mainly prove the following conjecture of Z.-H. Sun cite{SH20}: Let $p>3$ be a prime. Then $$\sum_{k=0}^{p-1}\binom{2k}k\frac{3k+1}{(-16)^k}f_k\equiv(-1)^{(p-1)/2}p+p^3E_{p-3}\pmod{p^4},$$ where $f_n=\sum_{k=0}^n\binom{n}k^3$ and $E_n$ stand for the $n$th Franel number and $n$th Euler number respectively.