**Fateh Mustapha Dehmeche, Douadi Mihoubi and Lahcene Ladjelat**

Notes on Number Theory and Discrete Mathematics

Print ISSN 1310–5132, Online ISSN 2367–8275

Volume 30, 2024, Number 1, Pages 163–169

DOI: 10.7546/nntdm.2024.30.1.163-169

**Full paper (PDF, 579 Kb)**

## Details

### Authors and affiliations

Fateh Mustapha Dehmeche

*Organisation Nationale pour les Innovateurs et la Recherche Scientifique
100 Rue de la Liberté, Boufarik, Algeria*

Douadi Mihoubi

* LMPA, University of M’sila, 28000 M’sila, Algeria*

Lahcene Ladjelat

* LMPA, University of M’sila, 28000 M’sila, Algeria*

### Abstract

In this paper, we give a characterization of primes and composite natural numbers using the notion of the sum of successive natural numbers. We prove essentially that an odd natural number is prime if and only if the unique decomposition of as a sum of successive natural numbers is the trivial decomposition with .

### Keywords

- Primes
- Composite numbers
- Sum of successive natural numbers
- Factorization
- Primality

### 2020 Mathematics Subject Classification

- 11A41
- 11A51
- 11Y05
- 11Y11

### References

- Cohen, H. (1966).
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*The Higher Arithmetic: An Introduction to Theory of Numbers*. Cambridge University Press, New York. - Guy, R. (1982). Sums of consecutive integers.
*The Fibonacci Quarterly*, 20, 36–38. - Kraft, J. S., Washington, L. C. (1990).
*An Introduction to Number Theory with Cryptography*. (2nd ed.). CRC Press A Chapman & Hall Book. - Mason, T. E. (1912). On the representation of an integer as the sum of consecutive integers.
*The American Mathematical Monthly*, 19, 46–50. - Pomerance, C. (2009).
*Computational Number Theory*. Princeton Companion to Mathematics Proof, Princeton University Press. - Prielipp, R. W., & Kuenzi, N. J. (1975). Sums of consecutive positive integers.
*The Mathematics Teacher*, 68, 18–21.

### Manuscript history

- Received: 9 May 2023
- Revised: 2 January 2024
- Accepted: 12 March 2024
- Online First: 12 March 2024

### Copyright information

Ⓒ 2024 by the Authors.

This is an Open Access paper distributed under the terms and conditions of the Creative Commons Attribution 4.0 International License (CC BY 4.0).

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## Cite this paper

Dehmeche, F. M., Mihoubi, D., & Ladjelat, L. (2024). Characterization of prime and composite numbers using the notion of successive sum of integers and the consequence in primality testing. *Notes on Number Theory and Discrete Mathematics*, 30(1), 163-169, DOI: 10.7546/nntdm.2024.30.1.163-169.