Supercongruences for the Almkvist-Zudilin numbers

TL;DR

This paper proves a conjecture on supercongruences for Almkvist-Zudilin numbers, revealing new divisibility properties and extending understanding of these special sequences in number theory.

Contribution

The paper establishes a proof for a conjecture on supercongruences specific to Almkvist-Zudilin numbers, advancing the theory of divisibility properties of these sequences.

Findings

Proved a conjecture on supercongruences for Almkvist-Zudilin numbers

Identified new divisibility properties of these sequences

Extended analysis to related sequence families

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 paper, we prove a conjecture on supercongruences for sequences that have come to be known as the Almkvist-Zudilin numbers. Some other (naturally) related family of sequences will be considered in a similar vain.