In this paper we show that, for any fixed 1<c<967/805, every sufficiently
large positive number N and a small constant Ξ΅>0, the diophantine
inequality \begin{equation*} |p_1^c+p_2^c+p_3^c+p_4^c-N|<\varepsilon
\end{equation*} has a solution in prime numbers p1β,p2β,p3β,p4β, such
that p1β=x2+y2+1.Comment: arXiv admin note: substantial text overlap with arXiv:2011.0396