Bertrand's postulate

Bertrand's postulate (also set by Bertrand - Chebyshev ) is a mathematical theorem, which states that there are n ≥ 1 is always a prime p between the number and twice the number is for natural numbers, so that: n < p ≤ 2n.

This claim was first erected in 1845 by the mathematician Joseph Bertrand, who proved for natural numbers to 3,000,000. The first complete proof for all natural numbers supplied Chebyshev five years later. Another, simpler proof was on the Indian mathematician Ramanujan S.. Furthermore, also led Paul Erdős in 1932 a simple proof.

Proof for n ≤ 4000

For the first 4000 natural numbers simply can specify primes, so the assertion holds. For the prime number sequence 2, 3, 5, 7, 13, 23, 43, 83, 163, 317, 631, 1259, 2503, 4001, a follower member is smaller than double the previous primes. Thus the assertion for n ≤ 4000 applies.