On a system of two Diophantine inequalities with six prime variables

TL;DR

This paper proves the solvability of a system of two Diophantine inequalities with six prime variables under certain conditions, improving previous results by Han-Liu-Zhang.

Contribution

It establishes new bounds for the solvability of a prime-based Diophantine system involving two inequalities with specific parameter constraints.

Findings

System solvable for sufficiently large N1, N2

Improves bounds on approximation errors in prime solutions

Extends previous work by Han-Liu-Zhang

Abstract

Suppose that $c,d,\alpha,\beta$ are real numbers satisfying the inequalities $1<d<c<79/71$ and $1<\alpha<\beta<6^{1-d/c}$. In this paper, it is proved that, for sufficiently large real numbers $N_1$ and $N_2$ subject to $\alpha\leqslant N_2/N_1^{d/c}\leqslant\beta$, the following Diophantine inequalities system \begin{align*} \begin{cases} |p_1^c+p_2^c+p_3^c+p_4^c+p_5^c+p_6^c-N_1|<\varepsilon_1 (N_1) \\ |p_1^d+p_2^d+p_3^d+p_4^d+p_5^d+p_6^d-N_2|<\varepsilon_2 (N_2) \end{cases} \end{align*} is solvable in prime variables $p_1, p_2, p_3, p_4, p_5, p_6$, where \begin{align*} \begin{cases} \varepsilon_1 (N_1)=N_1^{-(1/c)(79/71-c)} (\log N_1)^{201}, \\ \varepsilon_2 (N_2)=N_2^{-(1/d)(79/71-d)} (\log N_2)^{201} . \end{cases} \end{align*} This result constitutes an improvement upon the previous result of Han-Liu-Zhang [5].