Supercongruences

Supercongruences for polynomial analogs of the Apéry numbers

“…as conjectured by S. Chowla, J. Cowles and M. Cowles [12] and proved by I. Gessel [19] and Y. Mimura [26]. When n is divisible by p then this congruence can be further strengthened [8], [15]. Indeed, the congruence A(p r n) ≡ A(p r−1 n) holds modulo p 3r .…”

“…Note that, by the comments after (8), setting q = 1 in congruence (15) indeed recovers (14) for primes p ≥ 5. Our proof of Theorem 2.2 is a straightforward extension of the corresponding proof given in [33], where congruence (15) is proved in the case that n is a prime. We include the details of the proof in Appendix A for the benefit of the reader.…”

“…for all primes p ≥ 5 (in fact, a generalized version of these congruences already appears in M. Coster's thesis [15]). The congruences (2) are the special case (λ, µ) = (2, 2).…”

A p-adic analogue of a formula of Ramanujan

“…This term first appeared in the Ph.D. thesis of Coster [9] and refers to the fact that a congruence holds modulo p k for some k ≥ 2. Other examples of supercongruences have been observed in the context of number theory (see [22] and the references therein), mathematical physics [17], and algebraic geometry [26].…”

Divisibility properties of sporadic Apéry-like numbers

“…In his surprising proof [5,24] where n = n 0 + n 1 p + · · · + n r p r is the expansion of n in base p. Initial work of F. Beukers [8] and D. Zagier [29], which was extended by G. Almkvist, W. Zudilin [4] and S. Cooper [12], has complemented the Apéry numbers with a, conjecturally finite, set of sequences, known as Apéry-like, which share (or are believed to share) many of the remarkable properties of the Apéry numbers, such as connections to modular forms [2,7,27] or supercongruences [6,10,13,[21][22][23]. After briefly reviewing Apéry-like sequences in Section 2, we prove in Sections 3 and 4 our main result that all of these sequences also satisfy the Lucas congruences (1.2).…”