A note on Chebyshev’s theorem

A. Bërdëllima
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 31, 2025, Number 1, Pages 15–26
DOI: 10.7546/nntdm.2025.31.1.15-26
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Authors and affiliations

A. Bërdëllima
Faculty of Engineering, German International University in Berlin
Am Borsigturm 162, 13507, Berlin, Germany

Abstract

We revisit a classical theorem of Chebyshev about distribution of primes on intervals (n, 2n) n\in\mathbb N, and prove a generalization of it. Extending Erdős’ arithmetical-combinatorial argument, we show that for all k\in\mathbb N, there is n_k\in\mathbb N such that the intervals (kn, (k+1)n) contain a prime for all n\geq n_k. A quantitative lower bound is derived for the number of primes on such intervals. We also give numerical upper bounds for n_k for k \leq 20, and we draw comparisons with existing results in the literature.

Keywords

  • Bertrand’s postulate
  • Chebyshev’s theorem
  • Distribution of primes

2020 Mathematics Subject Classification

  • 11A41
  • 11-03

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Manuscript history

  • Received: 16 May 2024
  • Revised: 18 October 2024
  • Accepted: 26 March 2025
  • Online First: 28 March 2025

Copyright information

Ⓒ 2025 by the Author.
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

Bërdëllima, A. (2025). A note on Chebyshev’s theorem. Notes on Number Theory and Discrete Mathematics, 31(1), 15-26, DOI: 10.7546/nntdm.2025.31.1.15-26.

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