Bertrand's postulate
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- For Sylvester's theorem in plane geometry, see Sylvester–Gallai theorem.
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 [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:
where p ≤ x runs over primes, and the binomial coefficients. See proof of Bertrand's postulate for the details.
[edit] 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.
[edit] 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.)
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.
