Divisibility properties of sporadic Apéry-like numbers

  • Amita Malik
  • Armin Straub
Open Access


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.


Apéry-like numbers Lucas congruences p-adic properties 



This paper builds on experimental results obtained together with Arian Daneshvar, Pujan Dave and Zhefan Wang during an Illinois Geometry Lab (IGL) project during the Fall 2014 semester at the University of Illinois at Urbana-Champaign (UIUC). The aim of the IGL is to introduce undergraduate students to mathematical research. We wish to thank Arian, Pujan and Zhefan (at the time undergraduate students in engineering at UIUC) for their great work. In particular, their experiments predicted Corollaries 5.1 and 5.2, and provided the data for Table 4, which lead to Conjecture 6.3.

We are also grateful to Eric Rowland, who visited UIUC in October 2014, for interesting discussions on Apéry-like numbers and finite state automata, as well as for observing the congruence (6.1).

Moreover, we would like to express our gratitude to Tewodros Amdeberhan, Bruce C. Berndt, Robert Osburn and Wadim Zudilin for many helpful comments and encouragement. Finally, we thank the two referees for their detailed and helpful suggestions.


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Authors and Affiliations

  1. 1.Department of MathematicsUniversity of Illinois at Urbana-ChampaignUrbanaUSA
  2. 2.Department of Mathematics and StatisticsUniversity of South AlabamaMobileUSA

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