Some congruences on the hyper-sums of powers of integers involving Fermat quotient and Bernoulli numbers

Fouad Bounebirat and Mourad Rahmani
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 28, 2022, Number 3, Pages 533–541
DOI: 10.7546/nntdm.2022.28.3.533-541
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Authors and affiliations

Fouad Bounebirat
Department of Mathematics, University of Boumerdes
Boumerdes 35000, Algeria

Mourad Rahmani
Faculty of Mathematics, USTHB
P. O. Box 32, El Alia 16111, Bab-Ezzouar, Algiers, Algeria


For a given prime p ≥ 5, let ℤp denote the set of rational p-integers (those rational numbers whose denominator is not divisible by p). In this paper, we establish some congruences modulo a prime power p5 on the hyper-sums of powers of integers in terms of Fermat quotient, Wolstenholme quotient, Bernoulli and Euler numbers.


  • Bernoulli numbers
  • Congruence modulo a prime
  • Fermat quotient
  • Harmonic numbers
  • Wolstenholme quotient

2020 Mathematics Subject Classification

  • 11A07
  • 11B68
  • 11B83


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Manuscript history

  • Received: 24 March 2021
  • Revised: 5 August 2022
  • Accepted: 9 August 2022
  • Online First: 11 August 2022

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Cite this paper

Bounebirat, F., & Rahmani, M. (2022). Some congruences on the hyper-sums of powers of integers involving Fermat quotient and Bernoulli numbers. Notes on Number Theory and Discrete Mathematics, 28(3), 533-541, DOI: 10.7546/nntdm.2022.28.3.533-541.

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