Bertrand's Postulate

Bertrand's postulate (actually a theorem) states that for any integer n > 3, 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 . 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) used properties of the Gamma function to give a simpler proof, 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, defined as:

where p ≤ x runs over primes, and the binomial coefficients. See proof of Bertrand's postulate for the details.