Bertrand's postulate

Bertrand's postulate (also Bertrand-Chebyshev's theorem) is a mathematical theorem stating that for every natural number there exists at least one prime number with .

This assertion was first made in 1845 by the mathematician Joseph Bertrand, who proved it for natural numbers up to 3,000,000. The first complete proof for all natural numbers was given by Chebyshev five years later. Another, simpler proof was given by the Indian mathematician S. Ramanujan, who also introduced Ramanujan primes. In 1932, Paul Erdős also gave a simple proof.

Ramanujan proved a generalization, the existence of Ramanujan primes , such that for all between and there are at least primes.

Proof for n ≤ 4000

For the first 4000 natural numbers, prime numbers can easily be given, so the assertion holds. In the sequel

(sequence A295262 in OEIS)

of primes, each member of the sequence is less than twice the preceding member. Thus the assertion holds for