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

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.