Diophantine Inequalities as a Problem of Difference between Consecutive Primes

TL;DR

This paper introduces a novel method linking diophantine inequalities to the differences between consecutive primes, leading to proofs of existing theorems and new results, and discusses implications for prime conjectures.

Contribution

It develops a new approach connecting diophantine inequalities with prime gaps, enabling proofs of known theorems and deriving new results in number theory.

Findings

Proved Ingham's exponential theorem using the new method

Derived new results related to diophantine inequalities and prime gaps

Discussed implications for Cramer's and Andrica's conjectures

Abstract

In the present paper, we have developed a method for solving \textit{diophantine inequalities} using their relationship with the \textit{difference between consecutive primes}. Using this approach we have been able to prove some theorems, including Ingham's exponential theorem as well as some new results. Diophantine inequalities and their connection with Cramer's and Andrica's conjectures are also discussed.