On Proving of Diophantine Inequalities with Prime Numbers by Evaluations of the Difference between Consecutive Primes

TL;DR

This paper explores the proof of Diophantine inequalities involving prime numbers by evaluating the differences between consecutive primes, aiming to address longstanding conjectures.

Contribution

It provides detailed proofs of Legendre's and Oppermann's conjectures based on an evaluation of prime gaps under a specific hypothesis.

Findings

Proof of Legendre's conjecture

Proof of Oppermann's conjecture

Validation of prime gap hypothesis

Abstract

Using as the working hypothesis of an evaluation of the difference between primes $p_{n+1} - p_n = O(\sqrt{p_n})$ we represent in detail the proofs of Legendre's and Oppermann's conjectures.