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

Abstract

In this paper, we mainly prove the following conjecture of Z.-H. Sun cite{SH20}: Let $p>3$ be a prime. Then $$\sum _{k=0}^{p-1}\left(\genfrac{}{}{0ex}{}{2k}{k}\right)\frac{3k+1}{(-16{)}^k}{f}_k\equiv (-1{)}^{(p-1)/2}p+p^3{E}_{p-3}(\mathrm{mod}{p}^4),$$ where $f_n={\sum }_{k=0}^n{\left(\genfrac{}{}{0ex}{}{n}{k}\right)}^3$ and $E_n$ stand for the $n$th Franel number and $n$th Euler number respectively.

12 pages

References in corpus (1)

• On two congruences involving Apéry and Franel numbers