In the proof of the irrationality of $\zeta(3)$ and $\zeta(2)$, Ap\'ery defined two integer sequences through $3$-term recurrences, which are known as the famous Ap\'ery numbers. Zagier, Almkvist--Zudilin and Cooper successively introduced the other $13$ sporadic sequences through variants of Ap\'ery's $3$-term recurrences. All of the $15$ sporadic sequences are called Ap\'ery-like sequences. Motivated by Gessel's congruences mod $24$ for the Ap\'ery numbers, we investigate the congruences in the form $u_n\equiv \alpha^n \pmod{ N_{ \alpha}}~(\alpha\in \mathbb{ Z},N_{ \alpha}\in \mathbb{ N}^{ +})$ for all of the $15$ Ap\'ery-like sequences $\{ u_n\}_{ n\ge 0}$. Let $N_{ \alpha}$ be the largest positive integer such that $u_n\equiv \alpha^n \pmod{ N_{ \alpha}}$ for all non-negative integers $n$. We determine the values of $\max\{ N_{ \alpha}|\alpha \in \mathbb{ Z}\}$ for all of the $15$ Ap\'ery-like sequences $\{ u_n\}_{ n\ge 0}$.The binomial transforms of Ap\'ery-like sequences provide us a unified approach to this type of congruences for Ap\'ery-like sequences.
Categories: math.NT math.CO