On a diophantine inequality with prime numbers of a special type

Abstract

We consider the Diophantine inequality \[ \left| p_1^{c} + p_2^{c} + p_3^c- N \right| < (\log N)^{-E} , \] where $1<c<\frac{15}{14}$, $N$ is a sufficiently large real number and $E>0$ is an arbitrarily large constant. We prove that the above inequality has a solution in primes $p_1$, $p_2$, $p_3$ such that each of the numbers $p_1+2,p_2+2,p_3+2$ has at most $\left[\frac{369}{180-168c}\right]$ prime factors, counted with the multiplicity.

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