A system of two Diophantine inequalities with primes
Summary: Let \(1<d<c<128/119\), \(1<\alpha <\beta< 6^{1-d/c}\). In this paper, we prove that there exist positive real numbers \(N_1^{(0)}\) and \(N_2^{(0)}\) depending on \(c\), \(d\), \(\alpha\), \(\beta\) such that for all real numbers \(N_1>N_1^{(0)}\), \(N_2>N_2^{(0)}\) and \(\alpha\leq N_2/N_1^{d / c}\leq\beta\), the system of two Diophantine inequalities \(|p_1^c + \cdots + p_6^c - N_1|<N_1^{-(1/c) (128/119-c)} \log^{109} N_1,| p_1^d + \cdots + p_6^d - N_2|<N_2^{-(1/d) (128/119-d)} \log^{109} N_2\) is solvable in prime variables \(p_1,\dots, p_6\).
scientific article; zbMATH DE number 7362411
scientific article