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. The integers are commonly denoted by the above symbol. ... In mathematics, a prime number (or a prime) is a natural number that has exactly two (distinct) natural number divisors. ...

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 [1], from which the concept of Ramanujan primes would later arise, and Erdős (1913–1996) in 1932 published a simpler proof using the Chebyshev function θ(x), defined as: Joseph Louis François Bertrand (March 11, 1822 - April 5, 1900, born and died in Paris) was a French mathematician who worked in the fields of number theory, differential geometry, probability theory, and thermodynamics. ... Pafnuty Lvovich Chebyshev Pafnuty Lvovich Chebyshev (Russian: ) ( May 16 [O.S. May 4] 1821 – December 8 [O.S. November 26] 1894) was a Russian mathematician. ... Ramanujan Srinivasa Aiyangar Ramanujan (Tamil: &#3000;&#3021;&#2992;&#3008;&#2985;&#3007;&#2997;&#3006;&#3000; &#2960;&#2991;&#2969;&#3021;&#2965;&#3006;&#2992;&#3021; &#2992;&#3006;&#2990;&#3006;&#2985;&#3009;&#2972;&#2985;&#3021;) (December 22, 1887 &#8211; April 26, 1920) was a groundbreaking Indian mathematician. ... In mathematics, a Ramanujan prime is a prime number that satisfies a result proven by Srinivasa Ramanujan relating to the prime counting function. ... Paul ErdÅ‘s also Pál ErdÅ‘s, in English Paul Erdos or Paul Erdös, (March 26, 1913 – September 20, 1996) was an immensely prolific (and famously eccentric) Hungarian mathematician who, with hundreds of collaborators, worked on problems in combinatorics, graph theory, number theory, classical analysis, approximation theory, set... The Chebyshev function , with The Chebyshev function , for The Chebyshev function , for The Chebyshev function is either of two related functions. ...

where p ≤ x runs over primes, and the binomial coefficients. See proof of Bertrand's postulate for the details. In mathematics, particularly in combinatorics, the binomial coefficient of the natural number n and the integer k is the number of combinations that exist. ... In mathematics, Bertrands postulate states that for each n ≥ 2 there is a prime p such that n < p < 2n. ...

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. In mathematics, a permutation group is a group G whose elements are permutations of a given set M, and whose operation is the composition of permutations in G (which are thought of as bijective functions from the set M to itself); the relationship is often written as (G,M). ... James Joseph Sylvester James Joseph Sylvester (September 3, 1814 London - March 15, 1897 Oxford) was an English mathematician. ... Divisible is an Indie rock band from Los Angeles. ...

Erdős's theorems

Erdős proved 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.

Erdős also proved there always exists at least two prime numbers p with n < p < 2n for all n > 6. Moreover, one of them is congruent to 1 modulo 4, and another one is congruent to −1 modulo 4.

The prime number theorem (PNT) suggests that the number of primes between n and 2n is roughly n/ln(n) when n is large, and so in particular there are many more primes in this interval than are guaranteed by Bertrand's Postulate. That is, these theorems are comparatively weaker than the PNT. However, in order to use the PNT 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 PNT. 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 PNT and so has historical interest.) In number theory, the prime number theorem (PNT) describes the approximate, asymptotic distribution of the prime numbers. ...

A similar and still unsolved Legendre's conjecture asks whether for every n > 1, there is a prime p, such that n² < p < (n + 1)². Again we expect from the PNT 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 PNT are not (indeed, cannot be) sufficient to prove the existence of even one prime in this interval. Adrien-Marie Legendre conjectured that there is a prime number between n² and (n+1)² for every integer n > 0. ...

References

Erdos, P. "A Theorem of Sylvester and Schur." J. London Math. Soc. 9, 282-288, 1934.