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arXiv · 1508.00297

Divisibility properties of sporadic Apéry-like numbers

Amita Malik·Armin Straub
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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 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 $(η)$ 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.

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Amita Malik, Armin Straub. 2015-08-03. Divisibility properties of sporadic Apéry-like numbers. https://arxiv.org/abs/1508.00297

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