# Bertrand's postulate

Joseph Bertrand.

Bertrand's postulate, also known as **Bertrand's conjecture**, states that if *n* is an integer greater than 3, then there is at least one prime
number between *n* and 2*n*-2. This postulate (which should
now be called a theorem) is named after the French mathematician Joseph
Bertrand (1822–1900) who, in 1845, showed it was true for values of *n* up to 3,000,000. The Russian Pafnuty Chebyshev (1821–1894)
gave the first complete proof in 1850, so that it is sometimes called **Chebyshev's
theorem** (although another theorem also goes by this name). In 1932
Paul Erdös gave a more elegant proof,
using the binomial coefficients,
which is the one that appears in most modern textbooks. Bertrand's postulate
implies that the *n*-th prime *p*_{n} is
at most 2^{n}.