A system of two Diophantine inequalities with primes

Let k ≥ 7 be fixed, and v = (55k + 556)/(26k + 672),1 < d < c < v, 1 < μ < ν < k1−d/c and μ ≤ N2/N1d/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 |p1c + … + pkc − N1 | < N1−(1/c)(v−c) log195 N1, |p1d + … + pkd − N2 | < N2−(1/d)(v−d)log195 N2 has prime solutions p1,…,pk.