![cover image](https://wikiwandv2-19431.kxcdn.com/_next/image?url=https://upload.wikimedia.org/wikipedia/commons/thumb/9/9a/Bertrand.jpg/640px-Bertrand.jpg&w=640&q=50)
Bertrand's postulate
Existence of a prime number between any number and its double / From Wikipedia, the free encyclopedia
In number theory, Bertrand's postulate is the theorem that for any integer , there exists at least one prime number
with
![Thumb image](http://upload.wikimedia.org/wikipedia/commons/thumb/9/9a/Bertrand.jpg/320px-Bertrand.jpg)
A less restrictive formulation is: for every , there is always at least one prime
such that
Another formulation, where is the
-th prime, is: for
This statement was first conjectured in 1845 by Joseph Bertrand[2] (1822–1900). Bertrand himself verified his statement for all integers .
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 , the prime-counting function (number of primes less than or equal to
):