Divisibility properties of sporadic Apéry-like numbers

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 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.

Crouching AGM, Hidden Modularity

“…where n = n 0 + n 1 p + · · · + n r p r is the expansion of n in base p. In the case of Apérylike numbers, these congruences were shown in [18] and further general results were recently obtained in [1]. In addition, these sequences satisfy, or are conjectured to satisfy, the p ℓr -congruences…”

Lucas congruences for the Apéry numbers modulo $p^2$

“…, s j ). By expanding the product (8) to extract the coefficient of z m , one sees that this coefficient equals…”

“…, p − 1} supports a Lucas congruence for the sequence s(n) n∈Z modulo p α if s(d + pn) ≡ s(d)s(n) mod p α for all d ∈ D and for all n ∈ Z. As mentioned in the proof of Theorem 6, Malik and Straub[8, Lemma 6.2] proved that A(d) ≡ A(p − 1 − d) mod p for each d ∈ {0, 1, . .…”

Apéry-like numbers and families of newforms with complex multiplication

“…In general, integer sequences which are solutions to either of these recurrences are known as Apéry-like numbers. Congruence properties of many of these sequences have been studied by various authors [8,11,12,14,15,16,18,19].…”