On a binary Diophantine inequality involving primes of a special type

Abstract

Let $1<c<\frac{1787}{1502}$ and $N$ be a sufficiently large real number. In this paper, it is proved that for any arbitrarily large number $E>0$ and for almost all real $R\in (N,2N]$, the Diophantine inequality $$|p_1^c+p_2^c-R|<(logN{)}^{-E}$$ is solvable in prime variables $p_1,p_2$ such that, each of the numbers $p_1+2,p_2+2$ has at most $[\frac{79606}{35740-30040c}]$ prime factors, counted with multiplicity. Moreover, we prove that the Diophantine inequality $$|p_1^c+p_2^c+p_3^c+p_4^c-N|<(logN{)}^{-E}$$ is solvable in prime variables $p_1,p_2,p_3,p_4$ such that, each of the numbers $p_i+2(i=1,2,3,4)$ has at most $[\frac{93801402}{35740000-30040000c}]$ prime factors, counted with multiplicity.