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 p ≤ x 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