A simple proof of a curious congruence by Zhao

Abstract

Abstract. The author gives a simple proof of the following curious congru-

New congruences on multiple harmonic sums and Bernoulli numbers

“…This is the major obstacle in solving the cases for large number of variables. Nevertheless, from the process of determining R (2) n (p) and R (3) n (p) presented in this paper, we believe that for any integers n ≥ 3 and 1 ≤ m < n, there exist some rational numbers c a1,a2,...,a k and d a1,a2,...,a k such that for any prime p > n + 2 and r ≥ 2, we have For n ≤ 7, this has already been verified from Theorems 1 and 2 and the work of [4], [9] and [10]. For n ≥ 8, we are able to guess what the congruence should look like.…”

A Family of Supercongruences Involving Multiple Harmonic Sums

“…by Lemma 3.1 again. Now plugging (7), ( 8), ( 9) and ( 11) into (6), and then combining with (5), we get the desired result. Corollary 3.8.…”

“…This was proved by the last author using MHSs in [16], and by Ji using some combinatorial identities in [6]. Later on, a few generalizations and analogs were obtained by either increasing the number of indices and considering the corresponding supercongruences (see [11,13,15,21]), or considering the alternating version of MHSs (see [9,12]).…”

On a congruence involving harmonic series and Bernoulli numbers

“…Zhao's proof based on partial sums of a multiple zeta value series was published in [12] while Ji [4] presented an elementary proof slightly earlier. Meanwhile, this congruence can be refined along different directions.…”