Super congruences and Euler numbers

Abstract

Abstract. Let p > 3 be a prime. A p-adic congruence is called a super congruence if it happens to hold modulo some higher power of p. The topic of super congruences is related to many fields including Gauss and Jacobi sums and hypergeometric series. We prove that

“Divergent” Ramanujan-type supercongruences

“…The question mark indicates that the corresponding supercongruence remains conjectural. The non-questioned entries (1)-(3) are proved in this paper by extending the method of [19], while the supercongruence (4) (even in a more general form) is shown by Zhi-Wei Sun in his preprint [14].…”

“…After posting the preprint online we were informed by Zhi-Wei Sun that he had experimentally and independently discovered the congruences (1)- (8), but also proved in [14] a more general supercongruence than in (4). We thank him for bringing our attention to his work.…”

Symbolic summation methods and hypergeometric supercongruences

“…In [19], Sun also conjectured many congruences most of which have been confirmed. One of them is as follows: for any prime p > 3,…”

“…In 2011, Sun [19] investigated some congruences related to the Euler numbers. Especially, for any prime p > 3 he proved the following two congruences as extensions of (1.1):…”

Two triple binomial sum supercongruences

“…In recent literature, a variety of supercongruences have been conjectured by several people, such as Beukers [2], van Hamme [5], Rodriguez-Villegas [15], Zudilin [16], Chan et al [3], and lots more by Z.-W. Sun [11], [12]. Some of these conjectures are proved using a variety of methods, including the Gaussian hypergeometric series, the Wilf-Zeilberger method and p-adic analysis.…”