Bertrand's postulate

This article page is a stub, please help by expanding it.

---

Bertrand's postulate states that for any $n>1$, there is always at least one odd prime $p$ satisfying $n<p<2n$. Joseph Bertrand asserted this in 1845, but Pafnuty Chebyshev was the one who proved it in 1850. Decades later, Srinivasa Ramanujan came up with a short proof based on properties of the Gamma function. In 1932, an elementary proof was given by Paul Erdős which was based on the central binomial coefficients.

Bertrand's postulate does not prove the Goldbach conjecture, but it does suggest a counterexample, if it exists, would be very difficult to find. Given an even number and the available pairs of odd numbers that add up to it, Bertrand's postulate guarantees that at least one pair will have one prime, but it does not guarantee that there will be a pair with both numbers prime.

External links

• Eric W. Weisstein, Bertrand's Postulate, from MathWorld — A Wolfram Web Resource.
• Bertrand's conjecture, from PlanetMath Encyclopedia — Math for the people, by the people.