A short note on Erdös’ proof of Bertrand’s postulate
Year: 2026
Type: article
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.Mathematics Subject Classification (2020):11A41, 11A51
Source: Istanbul Journal of Mathematics
Author: Oliver Bültel
Institution: Boğaziçi University
Language: English
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: diamond
OpenAlex ID: W7166696734
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