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Divisibility properties of sporadic Apéry-like numbers

Research in Number Theory20162:5

  • Received: 3 August 2015
  • Accepted: 6 January 2016
  • Published:


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