On the Difference between Consecutive Primes and Estimates of the Number of Primes in the Interval $(n,2n)$

Abstract

Using evaluations of the difference between consecutive primes we develop another way of estimating of the number of primes in the interval $(n,2n)$. We also discuss the ultra Cramer conjecture, $p_{n+1}-p_n=O(lo{g}^{1+\epsilon }{p}_n)$ where $\epsilon >0$, in the context of the results we have obtained in our paper.