Proof of Bertrand's postulate

rdf:langString Proof of Bertrand's postulate

rdf:langString Bertrand's Postulate

rdf:langString BertrandsPostulate

rdf:langString In mathematics, Bertrand's postulate (actually a theorem) states that for each there is a prime such that . It was first proven by Chebyshev, and a short but advanced proof was given by Ramanujan. The following elementary proof was published by Paul Erdős in 1932, as one of his earliest mathematical publications. The basic idea is to show that the central binomial coefficients need to have a prime factor within the interval in order to be large enough. This is achieved through analysis of the factorization of the central binomial coefficients. The main steps of the proof are as follows. First, show that the contribution of every prime power factor in the prime decomposition of the central binomial coefficient is at most . Then show that every prime larger than appears at most once. The next step is to prove that has no prime factors in the interval . As a consequence of these bounds, the contribution to the size of coming from the prime factors that are at most grows asymptotically as for some . Since the asymptotic growth of the central binomial coefficient is at least , the conclusion is that, by contradiction and for large enough , the binomial coefficient must have another prime factor, which can only lie between and . The argument given is valid for all . The remaining values of are by direct inspection, which completes the proof.