Diophantine approximation with one prime of the form $p=x^2+y^2+1$

Abstract

Let $\epsilon >0$ be a small constant. In the present paper we prove that whenever $\eta$ is real and constants ${\lambda }_i$ satisfy some necessary conditions, then there exist infinitely many prime triples $p_1,p_2,p_3$ satisfying the inequality \begin{equation*} |λ_1p_1 + λ_2p_2 + λ_3p_3+η|<\varepsilon \end{equation*} and such that $p_3=x^2+y^2+1$.