Diophantine approximation with one prime of the form p = x2 + y2 + 1

Abstract

Let ε > 0 be a small constant. We prove that whenever η is real and constants λi satisfy some necessary conditions, then there exist infinitely many prime triples p1, p2, p3 satisfying the inequality |λ1p1+λ2p2+λ3p3+η| < ε and such that p3 = x2 + y2 + 1.