A Short Note on Erdös’ Proof of Bertrand’s Postulate
Abstract
We explain how a refinement of a method of Erdös leads to an inequality 1 < Î𝑛<𝑝≤2𝑛 𝑝 for all 𝑛 ≥ 9. The usual methods achieve this only for 𝑛 ≥ 4000, so that deducing Bertrand’s postulate still requires to check the primality of certain small numbers, typically 2, 3, 5, 7, 13, 23, 43, 83, 163, 317, 631, 1259, 2503 and 4001. In contrast, the only specific prime numbers that occur in our proof are 2, 3, 5, 7 and 11. For all 𝑛 ≥ 15 we also obtain the existence of at least four prime numbers between 𝑛 and 2𝑛, again with completely elementary means complemented by checking the primality of a few numbers, namely 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 79, 83, 89 and 97.
Keywords
References
- Erdös, P., 1932, Beweis eines Satzes von Tschebyschef, Acta Scientifica Mathematica 5, 194-198. google scholar
- Chandrasekharan, K., 1966, Einführung in die Analytische Zahlen-theorie, Lecture Notes in Mathematics vol. 29, Springer-Verlag, Berlin. google scholar
- Narkiewicz, W., 2000, The Development of Prime Number Theory, Springer-Verlag, New York. google scholar
Details
Primary Language
English
Subjects
Algebra and Number Theory
Journal Section
Research Article
Authors
Oliver Bültel
*
0000-0002-6370-8626
Türkiye
Publication Date
June 30, 2026
Submission Date
December 24, 2025
Acceptance Date
June 28, 2026
Published in Issue
Year 2026 Volume: 4 Number: 1