Bertrand's postulate

Bertrand's postulate In number theory, Bertrand's postulate is a theorem stating that for any integer {displaystyle n>3} , there always exists at least one prime number {displaystyle p} with {displaystyle n1} there is always at least one prime {displaystyle p} such that {displaystyle np_{i}{text{ for }}i>k{text{ where }}k=pi (p_{k})=pi (R_{n}),,} with pk the kth prime and Rn the nth Ramanujan prime.

Other generalizations of Bertrand's Postulate have been obtained using elementary methods. (In the following, n runs through the set of positive integers.) In 2006, M. El Bachraoui proved that there exists a prime between 2n and 3n.[5] In 1973, Denis Hanson proved that there exists a prime between 3n and 4n.[6] Furthermore, in 2011, Andy Loo proved that as n tends to infinity, the number of primes between 3n and 4n also goes to infinity,[7] thereby generalizing Erdős' and Ramanujan's results (see the section on Erdős' theorems below). The first result is obtained with elementary methods. The second one is based on analytic bounds for the factorial function.

Sylvester's theorem Bertrand's postulate was proposed for applications to permutation groups. Sylvester (1814–1897) generalized the weaker statement with the statement: the product of k consecutive integers greater than k is divisible by a prime greater than k. Bertrand's (weaker) postulate follows from this by taking k = n, and considering the k numbers n + 1, n + 2, up to and including n + k = 2n, where n > 1. According to Sylvester's generalization, one of these numbers has a prime factor greater than k. Since all these numbers are less than 2(k + 1), the number with a prime factor greater than k has only one prime factor, and thus is a prime. Note that 2n is not prime, and thus indeed we now know there exists a prime p with n < p < 2n. Erdős's theorems In 1932, Erdős (1913–1996) also published a simpler proof using binomial coefficients and the Chebyshev function θ, defined as: {displaystyle vartheta (x)=sum _{p=2}^{x}ln(p)} where p ≤ x runs over primes. See proof of Bertrand's postulate for the details.[8] Erdős proved in 1934 that for any positive integer k, there is a natural number N such that for all n > N, there are at least k primes between n and 2n. An equivalent statement had been proved in 1919 by Ramanujan (see Ramanujan prime).

Better results It follows from the prime number theorem that for any real {displaystyle varepsilon >0} there is a {displaystyle n_{0}>0} such that for all {displaystyle n>n_{0}} there is a prime {displaystyle p} such that {displaystyle n

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