Super congruences for two Apéry-like sequences

Abstract

In this paper we present many results and conjectures on congruences involving two types of Apéry-like sequences {G n (x)} and {V n (x)}.MSC: Primary 11A07, Secondary 05A19,11B65,11B68,11E25,65Q30

Supercongruences for sums involving Domb numbers

“…The second purpose of this paper is to prove a related supercongruence conjectured by Z.-H. Sun [20,Conjecture 2.6] and two divisibility results on sums of Domb numbers conjectured by Z.-W. Sun [23,Conjecture 77 (i)].…”

“…In the past few years, supercongruences for sums of Domb numbers have been widely discussed by many researchers (see, for example, [14,15,19,20,22,25]).…”

Super congruences concerning binomial coefficients and Apéry-like numbers

“…In Section 2 we show that for b p ≤ a p and b p ≤ p − 1 − a p , (1.5) ≡ 4x 2 − 2p (mod p 2 ) for p = x 2 + 4y 2 ≡ 1 (mod 4), where x and y are integers. We mention that in [S11,Theorem 2.8 and (6)] the author proved that…”

Congruences involving binomial coefficients and Apery-like numbers

“…For the congruences involving T n see the author's paper [S6]. For the formulas and congruences involving V n see [AZ,S8,S9,Su5,W,Z]. For the congruences concerning…”