Calculations for Bertrand's Postulate

Abstract

In 1845 J. Bertrand conjectured in [1] that for all n > 6, there is at least one prime between n/2 and n 2. Actually, Euler had already formulated this conjecture nearly a hundred years earlier (see [9]). He used. the form which now is well known as Bertrand's postulate, 7r(2 n) -g(n) > 0 for all n > 1, where r( n) denotes the number of primes less than or equal to n. The first proof was given by Tchebychev in [15], 369-382. Moreover, Tchebychev proved the following extension of Bertrand's postulate for e0 = 1/ 5: For any e > eo > 0, there exists an n (e) so that for all n >n(e),