Divisibility properties of sporadic Apéry-like numbers

被引：15

作者：

Malik A. ^{[1
]}

Straub A. ^{[1
,2
]}

机构：

[1] Department of Mathematics, University of Illinois at Urbana-Champaign, 1409 W Green St, Urbana, 61801, IL

[2] Department of Mathematics and Statistics, University of South Alabama, 411 University Blvd N, Mobile, 36688, AL

来源：

Research in Number Theory | / 2卷 / 1期

关键词：

Apéry-like numbers; Lucas congruences; p-adic properties;

D O I：

10.1007/s40993-016-0036-8

中图分类号：

学科分类号：

摘要：

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 s18 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. © 2016, The Author(s).

引用

共 45 条

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