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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 , where p runs over the primes.
Highlights
Several authors have discussed estimates for differences between consecutive primes
The proof uses the work done by Lowell Schoenfeld [3]
It is based on Theorem 7" from his paper which states that
Summary
Several authors (for example [1], [2]) have discussed estimates for differences between consecutive primes. The proof uses the work done by Lowell Schoenfeld [3]. It is based on Theorem 7" from his paper which states that. The importance of Theorem is the following. By setting 1/2 we get Bertrand’s Postulate This theorem is a generalization of this postulate. The importance of Theorem 2 is that it suggests that if p ’A :::1 a prime between x and x Kx" where K is a positive constant. Barkley Rosser and Lowell Schoenfeld [4] gtvc numerical evidence for the hypothesis of Theorem 2 in the case of p A. What we need is x(1 -.008/In(x)) rx(l + .(X)8/inCrx)) 0
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