Congruences for the Almkvist-Zudilin numbers

TL;DR

This paper investigates supercongruences for Almkvist-Zudilin numbers, establishing new divisibility properties and congruences that deepen understanding of their p-adic behavior.

Contribution

It proves new supercongruences for Almkvist-Zudilin numbers and related sequences, advancing knowledge of their divisibility properties.

Findings

Established supercongruences for Almkvist-Zudilin numbers

Identified relations between sequence values modulo prime powers

Enhanced understanding of p-adic properties of these sequences

Abstract

Given a prime number $p$, the study of divisibility properties of a sequence $c(n)$ has two contending approaches: $p$-adic valuations and superconcongruences. The former searches for the highest power of $p$ dividing $c(n)$, for each $n$; while the latter (essentially) focuses on the maximal powers $r$ and $t$ such that $c(p^rn)$ is congruent to $c(p^{r-1}n)$ modulo $p^t$. This is called supercongruence. In this note, we prove modest supercongruences for certain sequences that have come to be known as the Almkvist-Zudilin numbers and two other naturally related ones.