Bertrand's postulate

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Bertrand's postulate states that if n>3 is an integer, then there always exists at least one prime number p with n < p < 2n-2. A weaker but more elegant formulation is: for every 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 (1822-1900). Bertrand himself verified his statement for all numbers in the interval [2, 3 × 106]. His conjecture was completely proved by Pafnuty Lvovich Chebyshev (1821-1894) in 1850 and so the postulate is also called Chebyshev's theorem. Srinivasa Aaiyangar Ramanujan (1887-1920) gave a simpler proof and Paul Erdős (1913-1996) in 1932 published a simpler proof using the function θ(x), defined as:

<math> \theta(x) = \sum_{p=2}^{x} \ln (p) <math>

where px runs over primes, and the binomial coefficients. See proof of Bertrand's postulate for the details.

Sylvester's theorem

Bertrand's postulate was proposed for applications to permutation groups. James Joseph Sylvester (1814-1897) 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 asks whether for every n>1, there is a prime p, such that n2 < p < (n+1)2.fr:Postulat de Bertrand it:Postulato di Bertrand ru:Постулат Бертрана sl:Bertrandova domneva

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