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 Chebyshev (1821-1894) in 1850 and so the postulate is also called the Bertrand-Chebyshev theorem or Chebyshev's theorem. Ramanujan (1887-1920) gave a simpler proof and 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. 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.

Ramanujan's theorem

Ramanujan proved in 1919 that there always exists at least two prime number p with n < p < 2n except for a few small values of n (all less than 6); there always exists at least three primes in that interval except for a few more exceptions (the last is n=8); and more generally, he proved that for every k, the number of primes in that same interval is at least k except for a finite list of exceptions. (Obviously there are more exceptional values of n when k is larger.)

Erdős also proved there always exists at least two prime number p with n < p < 2n for all n > 6. Moreover, he showed that one of it must be of the form 4k+1 and other must be of the form 4k+3.

The prime number theorem suggests that the number of primes between n and 2n is roughly <math>\frac{n}{\ln n}</math> when n is large, and so in particular there are many more primes in this interval than are guaranteed by Bertrand's Postulate (or Ramanujan's theorem). That is, these theorems are comparatively weaker than the Prime Number Theorem. However, in order to use the Prime Number Theorem to prove results like Bertrand's Postulate, we would have to have very tight bounds on the error terms in the theorem -- that is, we have to know fairly precisely what "roughly" means in the Prime Number Theorem. Such error estimates are available but are very difficult to prove (and the estimates are only sufficient for large values of n). By contrast, Bertrand's Postulate can be stated more memorably and proved more easily, and makes precise claims about what happens for small values of n. (In addition, Chebyshev's theorem was proved before the Prime Number Theorem and so has historical interest.)

A similar and still unsolved conjecture asks whether for every n > 1, there is a prime p, such that n² < p < (n + 1)². Again we expect from the Prime Number Theorem that there will be not just one but many primes between n² and (n + 1)², but in this case the error estimates on the Prime Number Theorem are not (indeed, cannot be) sufficient to prove the existence of even one prime in this interval.fr:Postulat de Bertrand it:Postulato di Bertrand pl:Twierdzenie Czebyszewa ru:Постулат Бертрана sl:Bertrandova domneva sr:Бертрандов постулат ja:ベルトランの仮説 vi:Định đề Bertrand