On a ternary diophantine inequality with prime numbers of a special type II

Abstract

Suppose that N is a sufficiently large real number and E is an arbitrarily large constant. In this paper, it is proved that, for \(1< c < \frac{7}{6}\), the Diophantine inequality

$$\begin{aligned} |p_1^c+p_2^c+p_3^c-N|<(\log N)^{-E} \end{aligned}$$

is solvable in prime variables \(p_1,p_2,p_3\) so that each of the numbers \(p_i+2,\,i=1,2,3\), has at most \(\big [3.43655+{\frac{12.12}{7-6c}}\big ]\) prime factors counted with multiplicity.