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Mathematica Slovaca

Editor-in-Chief: Pulmannová, Sylvia

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Volume 68, Issue 5

Issues

Congruences involving alternating harmonic sums modulo pα qβ

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,kPp1ijk-2pr-1Bp-3(modpr),

where Pn 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=pαqβi,j,kP2pq1ijk78(2-q)(1-1q3)pα-1qβ-1Bp-3(modpα)

and

i+j+k=pαqβi,j,kPpq(-1)iijk12(q-2)(1-1q3)pα-1qβ-1Bp-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

About the article

Received: 2017-01-02

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, DOI: https://doi.org/10.1515/ms-2017-0159.

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