**Bertrand's postulate** states that if *n*>3 is an integer, then there always exists at least one prime number *p* with *n* < *p* < 2*n-2*. An equivalent weaker but more elegant formulation is: for every *n* > 1 there is always at least one prime *p* such that *n* < *p* < 2*n*.

This statement was first conjectured in 1845 by Joseph Bertrand (1822-1900). His conjecture was completely proved by Pafnuty Lvovich Chebyshev (1821-1894) in 1850 and so the postulate is also called Chebyshev's theorem. Chebyshev in his proof used Chebyshev's inequality. Bertrand himself verified his statement for all numbers in the interval [2, 3 × 10^{6}].

Srinivasa Aaiyangar Ramanujan (1887-1920) gave a simpler proof and Paul Erdös (1913-1996) in 1932 published a simpler proof using the function θ(*x*), defined as:

Bertrand's postulate was proposed for applications to permutation groups. James Joseph 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*.

A similar and still unsolved conjecture is asking whether for every *n*>1, there is a prime *p*, such that *n*^{2} < *p* < (*n*+1)^{2}.