New Congruences on Multiple Harmonic Sums and Bernoulli Numbers

2015/04/13 by Liuquan Wang, Wang, Liuquan
Mathematics · #11A41 #FOS: Mathematics #Number Theory (math.NT) #Primary 11A07 #math.NT #msc:11A07 #msc:11A41

paper · pdf · doi:10.48550/arxiv.1504.03227

14 pages. This version simplifies the previous version. Moreover, two important theorems and a conjecture were added

arxiv created 2016/01/27 · arxiv updated 2016/01/28

Abstract

Let P_n denote the set of positive integers which are prime to n. Let B_n be the n-th Bernoulli number. For any prime p ≥ 11 and integer r≥ 2, we prove that ∑_\beginsmallmatrix l₁+l₂+⋯ +l₆=p^r l₁,⋯ ,l₆∈ P_p \endsmallmatrix\frac1l₁l₂l₃l₄l₅l₆≡ - (5!)/(18)p^r-1B_p-3² \pmodp^r. This extends a family of curious congruences. We also obtain other interesting congruences involving multiple harmonic sums and Bernoulli numbers.

Related

• A New Super Congruence Involving Multiple Harmonic Sums
• Some new curious congruences involving multiple harmonic sums
• A Curious Congruence Involving Alternating Harmonic Sums
• On a kind of generalized multi-harmonic sums
• On a kind of generilized multi-harmonic sum