Show Summary Details
More options …

# Mathematica Slovaca

Editor-in-Chief: Pulmannová, Sylvia

6 Issues per year

IMPACT FACTOR 2017: 0.314
5-year IMPACT FACTOR: 0.462

CiteScore 2017: 0.46

SCImago Journal Rank (SJR) 2017: 0.339
Source Normalized Impact per Paper (SNIP) 2017: 0.845

Mathematical Citation Quotient (MCQ) 2017: 0.26

Online
ISSN
1337-2211
See all formats and pricing
More options …
Volume 68, Issue 5

# Congruences involving alternating harmonic sums modulo pαqβ

Zhongyan Shen
/ Tianxin Cai
Published Online: 2018-10-20 | DOI: https://doi.org/10.1515/ms-2017-0159

## Abstract

In 2014, Wang and Cai established the following harmonic congruence for any odd prime p and positive integer r,

$∑i+j+k=pri,j,k∈Pp1i⁢j⁢k≡-2⁢pr-1⁢Bp-3(modpr),$

where ${\mathcal{P}}_{n}$ denote the set of positive integers which are prime to n.

In this note, we obtain the congruences for distinct odd primes p, q and positive integers α, β,

$∑i+j+k=pα⁢qβi,j,k∈P2⁢p⁢q1i⁢j⁢k≡78⁢(2-q)⁢(1-1q3)⁢pα-1⁢qβ-1⁢Bp-3(modpα)$

and

$∑i+j+k=pα⁢qβi,j,k∈Pp⁢q(-1)ii⁢j⁢k≡12⁢(q-2)⁢(1-1q3)⁢pα-1⁢qβ-1⁢Bp-3(modpα).$

MSC 2010: Primary 11A07, 11A41

Keywords: Bernoulli numbers; harmonic sums; congruences

This work is supported by the Natural Science Foundation of Zhejiang Province, Project (No. LY18A010016) and the National Natural Science Foundation of China, Project (No. 11571303)

## References

• [1]

CAI, T.—SHEN, Z.—JIA, L.: A congruence involving harmonic sums modulo pα qβ, Int. J. Number Theory 13 (2017), 1083–1094.Google Scholar

• [2]

JI, C.: A simple proof of a curious congruence by Zhao, Proc. Amer. Math. Soc. 133 (2005), 3469–3472.Google Scholar

• [3]

WANG, L.—CAI, T. : A curious congruence modulo prime powers, J. Number Theory 144 (2014), 15–24.Google Scholar

• [4]

XIA, B.—CAI, T.: Bernoulli numbers and congruences for harmonic sums, Int. J. Number Theory 6 (2010), 849–855.Google Scholar

• [5]

ZHAO, J.: Congruences involving multiple harmonic sums and finite multiple zeta values, arxiv:1404.3549.Google Scholar

• [6]

ZHAO, J.: Bernoulli numbers, Wolstenholme’s theorem, and p5 variations of Lucas’ theorem, J. Number Theory 123 (2007), 18–26.Google Scholar

Accepted: 2017-05-30

Published Online: 2018-10-20

Published in Print: 2018-10-25

Communicated by Federico Pellarin

Citation Information: Mathematica Slovaca, Volume 68, Issue 5, Pages 975–980, ISSN (Online) 1337-2211, ISSN (Print) 0139-9918,

Export Citation