A system of two Diophantine inequalities with primes

School of Mathematics and Statistics, Shandong Normal University, Jinan 250014, Shandong, PR China

^* Corresponding author: This email address is being protected from spambots. You need JavaScript enabled to view it.

Received: 4 November 2025
Accepted: 31 December 2025

Abstract

Let k ≥ 7 be fixed, and v = (55k + 556)/(26k + 672),1 < d < c < v, 1 < μ < ν < k^1−d/c and μ ≤ N₂/N₁^d/c ≤ v. Suppose that N^′₁, N^′₂ > 0 are real numbers, and that N₁, N₂ are real numbers satisfying N₁ > N^′₁ and N₂ > N^′₂. Then we prove the system of two Diophantine inequalities

|p₁^c + … + p_k^c − N₁ | < N₁^{−(1/c)(v−c)} log¹⁹⁵ N₁,

|p₁^d + … + p_k^d − N₂ | < N₂^{−(1/d)(v−d)}log¹⁹⁵ N₂

has prime solutions p₁,…,p_k.