Divisibility properties of sporadic Apéry-like numbers

Abstract

In 1982, Gessel showed that the Apéry numbers associated to the irrationality of $\zeta (3)$ satisfy Lucas congruences. Our main result is to prove corresponding congruences for all sporadic Apéry-like sequences. In several cases, we are able to employ approaches due to McIntosh, Samol--van Straten and Rowland--Yassawi to establish these congruences. However, for the sequences often labeled $s_{18}$ and $(\eta )$ we require a finer analysis. As an application, we investigate modulo which numbers these sequences are periodic. In particular, we show that the Almkvist--Zudilin numbers are periodic modulo $8$, a special property which they share with the Apéry numbers. We also investigate primes which do not divide any term of a given Apéry-like sequence.

30 pages, 1 figure

References in corpus (1)

• Supercongruences for sporadic sequences