Divisibility properties of sporadic Apéry-like numbers

Divisibility properties of sporadic Apéry-like numbers
Amita Malik, Armin Straub — Research in Number Theory — Volume 2, Number 1, 2016, Pages 1-26, #5


In 1982, Gessel showed that the Apéry numbers associated to the irrationality of \(\zeta(3)\) 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 \((\eta)\) 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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    author = {Amita Malik and Armin Straub},
    title = {Divisibility properties of sporadic {A}p\'ery-like numbers},
    journal = {Research in Number Theory},
    year = {2016},
    volume = {2},
    number = {1},
    pages = {1--26, #5},
    doi = {10.1007/s40993-016-0036-8},