Open Access

Divisibility properties of sporadic Apéry-like numbers

Research in Number Theory20162:5

DOI: 10.1007/s40993-016-0036-8

Received: 3 August 2015

Accepted: 6 January 2016

Published: 14 February 2016

Abstract

In 1982, Gessel showed that the Apéry numbers associated to the irrationality of ζ(3) satisfy Lucas congruences. Our main result is to prove corresponding congruences for all known 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 labeled s 18 and (η) 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.

Keywords

Apéry-like numbers Lucas congruences p-adic properties