Bertrand's postulate

In number theory, Bertrand's postulate is the theorem that for any integer ${\displaystyle n>3}$, there exists at least one prime number ${\displaystyle p}$ with

${\displaystyle n<p<2n-2.}$

A less restrictive formulation is: for every ${\displaystyle n>1}$, there is always at least one prime ${\displaystyle p}$ such that

${\displaystyle n<p<2n.}$

Another formulation, where ${\displaystyle p_{n}}$ is the ${\displaystyle n}$-th prime, is: for ${\displaystyle n\geq 1}$

${\displaystyle p_{n+1}<2p_{n}.}$^[1]

This statement was first conjectured in 1845 by Joseph Bertrand^[2] (1822–1900). Bertrand himself verified his statement for all integers ${\displaystyle 2\leq n\leq 3\,000\,000}$.

His conjecture was completely proved by Chebyshev (1821–1894) in 1852^[3] and so the postulate is also called the Bertrand–Chebyshev theorem or Chebyshev's theorem. Chebyshev's theorem can also be stated as a relationship with ${\displaystyle \pi (x)}$, the prime-counting function (number of primes less than or equal to ${\displaystyle x}$):

${\displaystyle \pi (x)-\pi {\bigl (}{\tfrac {x}{2}}{\bigr )}\geq 1,{\text{ for all }}x\geq 2.}$