  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.
