Bertrand's postulate
Existence of a prime number between any number and its double / From Wikipedia, the free encyclopedia
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In number theory, Bertrand's postulate is the theorem that for any integer , there exists at least one prime number with
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 ):