Congruences for the Ap\'ery numbers modulo $p^3$

TL;DR

This paper establishes new congruences modulo p^3 for Apéry numbers and related sequences, revealing intricate number-theoretic properties and extending known results for primes congruent to 3 mod 4.

Contribution

It provides explicit congruences for Apéry numbers and a sequence defined by a recurrence, advancing understanding of their behavior modulo prime powers.

Findings

A'_{(p-1)/2} mod p^3 for primes p ≡ 3 mod 4

Congruences for t_p, t_{p-1}, and t_{(p-1)/2} modulo p^3 or p^2

New relations linking Apéry numbers and recurrence-defined sequences modulo prime powers

Abstract

Let $\{A'_n\}$ be the Ap\'ery numbers given by $A'_n=\sum_{k=0}^n\binom nk^2\binom{n+k}k.$ For any prime $p\equiv 3\pmod 4$ we show that $A'_{\frac{p-1}2}\equiv \frac{p^2}3\binom{\frac{p-3}2}{\frac{p-3}4}^{-2}\pmod {p^3}$. Let $\{t_n\}$ be given by $$t_0=1,\ t_1=5\quad\hbox{and}\quad t_{n+1}=(8n^2+12n+5)t_n-4n^2(2n+1)^2t_{n-1}\ (n\ge 1).$$ We also obtain the congruences for $t_p\pmod {p^3},\ t_{p-1}\pmod {p^2}$ and $t_{\frac{p-1}2}\pmod {p^2}$, where $p$ is an odd prime.