Diophantine Inequalities with Primes, Auxiliary Inequalities, Evaluations of the Difference between Consecutive Primes

TL;DR

This paper introduces a new method for proving Diophantine inequalities involving primes using auxiliary inequalities and prime gap evaluations, providing new proofs and results on primes in specific intervals.

Contribution

It presents a novel approach to Diophantine inequalities with primes, including an alternative proof of Ingham's theorem with computable constants and results on primes in various intervals.

Findings

Alternative proof of Ingham's theorem with effective constants

Results on primes in intervals of the form ((n-1)^k, n^k)

Results on primes in intervals ((k-1)/k n, (k/k-1) n)

Abstract

The goal of the present paper is to present a method of proving of Diophantine inequalities with primes through the use of auxiliary inequalities and available evaluations of the difference between consecutive primes. We study the Legendre - Ingham's problem on primes in intervals $((n - 1)^k, n^k)$ and also a problem on primes in intervals $(\frac{k-1}{k}n, \frac{k}{k-1}n)$ when $k$ is a real number. A number of the new results including an alternative proof of Ingham's theorem with the effectively computable constant and also Ingham's theorem with two primes are proved.