A quinary diophantine inequality by primes with one of the form $\mathbf{p}={\mathbf{x}}^2+{\mathbf{y}}^2+1$

Abstract

In this paper we show that, for any fixed $1<c<\frac{5363}{3900}$, every sufficiently large positive number $N$ and a small constant $\epsilon >0$, the diophantine inequality \begin{equation*} |p_1^c+p_2^c+p_3^c+p_4^c+p_5^c-N|<\varepsilon \end{equation*} has a solution in prime numbers $p_1,p_2,p_3,p_4,p_5$, such that $p_1=x^2+y^2+1$.

arXiv admin note: substantial text overlap with arXiv:2012.06476, arXiv:2011.03967

References in corpus (1)

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