# Congruences for Ap\'ery numbers $\beta_{n}=\sum_{k=0}^{n}\binom{n}{k}^2\binom{n+k}{k}$

## TL;DR

This paper investigates new congruences involving Apéry numbers, revealing deep number-theoretic properties and relations between different types of Apéry sequences, with implications for understanding their modular behavior.

## Contribution

The paper establishes novel congruences for Apéry numbers and explores their relations, advancing the understanding of their arithmetic properties.

## Key findings

- Proved a congruence involving Apéry numbers modulo 2n^2.
- Derived a prime modulus congruence involving Bernoulli numbers.
- Identified relations between two types of Apéry numbers.

## Abstract

In this paper we establish some congruences involving the Ap\'ery numbers $\beta_{n}=\sum_{k=0}^{n}\binom{n}{k}^2\binom{n+k}{k}$ $(n=0,1,2,\ldots)$. For example, we show that $$\sum_{k=0}^{n-1}(11k^2+13k+4)\beta_k\equiv0\pmod{2n^2}$$ for any positive integer $n$, and $$\sum_{k=0}^{p-1}(11k^2+13k+4)\beta_k\equiv 4p^2+4p^7B_{p-5}\pmod{p^8}$$ for any prime $p>3$, where $B_{p-5}$ is the $(p-5)$th Bernoulli number. We also present certain relations between congruence properties of the two kinds of A\'pery numbers, $\beta_n$ and $A_n=\sum_{k=0}^n\binom nk^2\binom{n+k}k^2$.