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Licensed Unlicensed Requires Authentication Published by De Gruyter October 20, 2018

Congruences involving alternating harmonic sums modulo pαqβ

Zhongyan Shen and Tianxin Cai
From the journal Mathematica Slovaca

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).

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)


  1. Communicated by Federico Pellarin

Acknowledgement

The authors would like to thank the referee for his/her valuable comments and suggestions, and also the authors thank the China Scholarship Council for supporting our research.

References

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Received: 2017-01-02
Accepted: 2017-05-30
Published Online: 2018-10-20
Published in Print: 2018-10-25

© 2018 Mathematical Institute Slovak Academy of Sciences