Bertrand's Postulate - Better Results

Better Results

It follows from the prime number theorem that for any real k > 1, there exists an n₀ such that there is always a prime between n and kn for all n > n₀: it can be shown, for instance, that

which implies that π(kn) − π(n) goes to infinity (and in particular is greater than 1 for sufficiently large n).

Non-asymptotic bounds have been also been proved. In 1952, Jitsuro Nagura proved that for n ≥ 25, there is always a prime between n and (1 + 1⁄₅)n.

In 1976, Lowell Schoenfeld showed that for n ≥ 2010760, there is always a prime between n and (1 + 1⁄₁₆₅₉₇)n. In 1998, Pierre Dusart improved the result in his doctoral thesis, showing that for k ≥ 463, p_{k + 1} ≤ (1 + 1⁄_{(ln2 pk)})p_k, and in particular for x ≥ 3275, there exists a prime number between x and (1 + 1⁄_(2ln2x))x. In 2010 he proved, that for x ≥ 396738 there is at least one prime between x and (1 + 1⁄_(25ln2x))x.

Generalizations of Bertrand's Postulate have also been obtained by elementary methods. (In the following, n runs through the set of positive integers.) In 2006, M. El Bachraoui proved that there exists a prime between 2n and 3n. In 2011, Andy Loo proved that there exists a prime between 3n and 4n. Furthermore, he proved that as n tends to infinity, the number of primes between 3n and 4n also goes to infinity, thereby generalizing Erdős' and Ramanujan's results (see the section on Erdős' theorems above). None of these proofs require the use of deep analytic results.