Bertrand’s Postulate

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Chapter 1 Bertrand’s Postulate

Bertrand’s postulate asserts that for every $n \geq 1$ there is a prime $p$ such that $n \lt p \leq 2n\text{.}$ Our presentation mostly follows notes by David Galvin [1].

This statement was conjectured in 1845 by Joseph Bertrand, who verified it for $n \lt 3 \cdot 10^6\text{.}$ In 1850, Tchebychev proved it, using analysis. In 1919, Ramanujan gave a shorter proof, still using analysis. In 1932, in one of his first publications (perhaps his very first), Erdős gave an “elementary” proof. We will show Erdős’s proof, and then a fun corollary.

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