The quaternary Piatetski-Shapiro inequality with one prime of the form $\mathbf{p=x^2+y^2+1}$

TL;DR

This paper proves that for certain exponents, large numbers can be approximated by sums of four prime powers, with one prime of the form x^2 + y^2 + 1, extending the understanding of prime representations in Diophantine inequalities.

Contribution

It establishes the existence of solutions to a quaternary Piatetski-Shapiro inequality with a prime of the specific form x^2 + y^2 + 1 for a range of exponents c.

Findings

Solutions exist for large N within the specified range of c.

At least one prime in the sum can be represented as x^2 + y^2 + 1.

The result extends previous work on prime representations in Diophantine inequalities.

Abstract

In this paper we show that, for any fixed $1<c<967/805$, every sufficiently large positive number $N$ and a small constant $\varepsilon>0$, 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 $p_1,\,p_2,\,p_3,\,p_4$, such that $p_1=x^2 + y^2 +1$.