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

TL;DR

This paper introduces a new method for estimating the number of primes in the interval (n, 2n) based on differences between consecutive primes and discusses the ultra Cramer conjecture within this context.

Contribution

The paper presents a novel approach to estimate prime counts in (n, 2n) using prime gaps and explores implications for the ultra Cramer conjecture.

Findings

New estimation method for primes in (n, 2n)

Analysis of prime gap behavior related to conjectures

Discussion on implications for ultra Cramer conjecture

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.