Bertrand's postulate states that if n is a positive integer, then for n > 3 there always exists at least one prime number p between n and 2n2, or in an equivalent weaker but more elegant form then for n > 1 there is always at least one prime p such that n < p < 2n.
This statement was first conjectured in 1845 by Joseph Bertrand[?] (18221900). His conjecture was completely proved by Pafnuty Lvovich Chebyshev (18211894) in 1850 and so the postulate is also called Chebyshev's theorem. Chebyshev in his proof used the Chebyshev's inequality. Bertrand himself verified his statement for all numbers in the interval [2, 3 × 10^{6}].
Srinivasa Aaiyangar Ramanujan (18871920) gave a simpler proof and Paul Erdös (19131996) in 1932 published a very simple proof where he used the function θ(x), defined as:
where p ≤ x runs over primes, and the binomial coefficients.
Bertrand's postulate was proposed for applications to permutation groups. James Joseph Sylvester (18141897) generalized it with the statement: the product of k consecutive integers greater than k is divisible by a prime greater than k.
A similar and still unsolved conjecture is asking for a prime p, such that n^{2} < p < (n+1)^{2}.
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