A System of Two Diophantine Inequalities with Primes

Abstract

Let $1<d<c<128/119$ , $1<\alpha <\beta <6^{1-d/c}$ . In this paper, we prove that there exist positive real numbers $N_1^{\left(0\right)}$ and $N_2^{\left(0\right)}$ depending on $c,d,\alpha ,\beta$ such that for all real numbers $N_1>N_1^{\left(0\right)}$ , $N_2>N_2^{\left(0\right)}$ and $\alpha \le N_2/N_1^{d/c}\le \beta$ , the system of two Diophantine inequalities $\left|p_1^c+\cdots +p_6^c-N_1\right|<N_1^{-\left(1/c\right)\left(128/119-c\right)}{\log }^{109}{N}_1,\left|p_1^d+\cdots +p_6^d-N_2\right|<N_2^{-\left(1/d\right)\left(128/119-d\right)}{\log }^{109}{N}_2$ is solvable in prime variables $p_1,\dots ,p_6$ .