2024/06/26 by Liu, Ji-Cai
#05A10 #11B50 #11B65 #Combinatorics (math.CO) #FOS: Mathematics #Number Theory (math.NT)
paper · doi:10.48550/arxiv.2406.18059
In the proof of the irrationality of ζ(3) and ζ(2), Apéry defined two integer sequences through 3-term recurrences, which are known as the famous Apéry numbers. Zagier, Almkvist--Zudilin and Cooper successively introduced the other 13 sporadic sequences through variants of Apéry's 3-term recurrences. All of the 15 sporadic sequences are called Apéry-like sequences. Motivated by Gessel's congruences mod 24 for the Apéry numbers, we investigate the congruences in the form un≡ αn \pmodNα~(α∈ ℤ,Nα∈ ℕ+) for all of the 15 Apéry-like sequences \un\n≥ 0. Let Nα be the largest positive integer such that un≡ αn \pmodNα for all non-negative integers n. We determine the values of max\Nα|α∈ ℤ\ for all of the 15 Apéry-like sequences \un\n≥ 0.The binomial transforms of Apéry-like sequences provide us a unified approach to this type of congruences for Apéry-like sequences.