On congruences involving Ap\'ery numbers

TL;DR

This paper proves a new p-adic congruence involving Apéry numbers and sums of powers, confirming a conjecture by Z.-W. Sun for primes greater than 3.

Contribution

It establishes a p-adic congruence for sums involving Apéry numbers, confirming a conjecture and providing explicit dependence on a p-adic integer.

Findings

Proved a congruence for sums involving Apéry numbers modulo p^3.

Confirmed Z.-W. Sun's conjecture for all primes p > 3.

Identified a p-adic integer c_m depending only on m.

Abstract

In this paper, we mainly establish a congruence for a sum involving Ap\'{e}ry numbers, which was conjectured by Z.-W. Sun. Namely, for any prime $p>3$ and positive odd integer $m$, we prove that there is a $p$-adic integer $c_m$ only depending on $m$ such that $$\sum_{k=0}^{p-1}(2k+1)^{m}(-1)^kA_k\equiv c_mp\left(\frac{p}{3}\right)\pmod{p^3},$$ where $A_k=\sum_{j=0}^{k}\binom{k}{j}^2\binom{k+j}{j}^2$ is the Ap\'{e}ry number and $(\frac{\cdot}{p})$ is the Legendre symbol.