Proof of Bertrand's Postulate

Proof Of Bertrand's Postulate

In mathematics, Bertrand's postulate (actually a theorem) states that for each there is a prime such that . It was first proven by Pafnuty Chebyshev, and a short but advanced proof was given by Srinivasa Ramanujan. The gist of the following elementary proof is due to Paul Erdős. The basic idea of the proof is to show that a certain central binomial coefficient needs to have a prime factor within the desired interval in order to be large enough. This is made possible by a careful analysis of the prime factorization of central binomial coefficients.

The main steps of the proof are as follows. First, one shows that every prime power factor that enters into the prime decomposition of the central binomial coefficient is at most . In particular, every prime larger than can enter at most once into this decomposition; that is, its exponent is at most one. The next step is to prove that has no prime factors at all in the gap interval . As a consequence of these two bounds, the contribution to the size of coming from all the prime factors that are at most grows asymptotically as for some . Since the asymptotic growth of the central binomial coefficient is at least, one concludes that for large enough the binomial coefficient must have another prime factor, which can only lie between and . Indeed, making these estimates quantitative, one obtains that this argument is valid for all . The remaining smaller values of are easily settled by direct inspection, completing the proof of the Bertrand's postulate.