Diophantine approximation by prime numbers of a special form

S. I. Dimitrov, T. L. Todorova

Published 2016-10-13Version 1

We show that for $B>1$ and for 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 $|\lambda _1p_1 + \lambda _2p_2 + \lambda _3p_3+\eta| < [\log (\max p_j)]^{-B}$ and such that $p_i+2=P_8,i=1,2,3$. The proof uses Davenport - Heilbronn adaption of the circle method together with a vector sieve method.