DIOPHANTINE APPROXIMATION BY PRIME NUMBERS OF A SPECIAL FORM

Abstract

We show that for $B>1$ and for some constants ${\lambda }_i,i=1,2,3$ subject to certain assumptions, there are infinitely many prime triples $p_1,p_2,p_3$ satisfying the inequality $\mid {\lambda }_1{p}_1+{\lambda }_2{p}_2+{\lambda }_3{p}_3+\eta \mid <[log(max{p}_j){]}^{-B}$ and such that $p_1+2,p_2+2and{p}_3+2$ have no more than 8 prime factors. The proof uses Davenport - Heilbronn adaption of the circle method together with a vector sieve method.