A short note on Erdös’ proof of Bertrand’s postulate

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