International Journal of Mathematics and Mathematical Sciences
Volume 10 (1987), Issue 4, Pages 821-823
doi:10.1155/S0161171287000917
The generalization and proof of Bertrand's postulate
Department of Mathematics Physics and Computer Science, Ryerson Polytechnical Institute, Toronto M5B 2K3, Ontario, Canada
Received 21 May 1985; Revised 16 December 1986
Copyright © 1987 George Giordano. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract
The purpose of this paper is to show that for 0<r<1 one can determine explicitly an x0 such that ∀x≥x0, ∃ at least one prime between rx and x. This is a generalization of Bertrand's Postulate. Furthermore, the same procedures are used to show that if one can find upper and lower bounds for θ(x) whose difference is kxρ then ∃ a prime between x and x−Kxρ, where k, K>0 are constants, 0<ρ<1 and θ(x)=∑p≤xlnp, where p runs over the primes.