**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

**Full paper (PDF, 174 Kb)**

## Details

### 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
*

### Abstract

For a given prime *p* ≥ 5, let ℤ_{p} 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 *p*^{5} on the hyper-sums of powers of integers in terms of Fermat quotient, Wolstenholme quotient, Bernoulli and Euler numbers.

### Keywords

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

### 2020 Mathematics Subject Classification

- 11A07
- 11B68
- 11B83

### References

- Glaisher, J. W. L. (1901). On the residues of the sums of the inverse powers of numbers in arithmetical progression.
*Quarterly Journal of Mathematics*, 32, 271–288. - Graham, R. L., Knuth D. E., & Patashnik O. (1994).
*Concrete Mathematics*.

Addison-Wesley Publishing Company, Reading, MA. - Hardy, G. H., & Wright E. M. (1979).
*An Introduction to the Theory of Numbers*. Clarendon Press. - Jacobson, N. (1985).
*Basic Algebra I*. Second edition. W. H. Freeman and Company, New York. - Laissaoui, D., Bounebirat, F., & Rahmani, M. (2017). On the hyper-sums of powers of integers.
*Miskolc Mathematical Notes*, 18, 307–314. - McIntosh, R. J. (1995). On the converse of Wolstenholme’s Theorem.
*Acta Arithmetica*, 71, 381–389. - Sun, Z-H. (2000). Congruences concerning Bernolli numbers and Bernoulli polynomials.
*Discrete Applied Mathematics*, 105, 193–223. - Sun, Z-H. (2008). Congruences involving Bernoulli and Euler numbers.
*Journal of Number Theory*, 128, 280–312.

### Manuscript history

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

## Related papers

## 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.