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, $p_{n+1} - p_n = O(log^{1+\epsilon}p_n)$ where $\epsilon > 0$, in the context of the results we have obtained in our paper.