On Two Diophantine Inequalities Over Primes

Abstract

Let $1<c<37/18,c\ne 2$ and $N$ be a sufficiently large real number. In this paper, we prove that, for almost all $R\in (N,2N],$ the Diophantine inequality $|p_1^c+p_2^c+p_3^c-R|<{\log }^{-1}N$ is solvable in primes $p_1,p_2,p_3.$ Moreover, we also investigate the problem of six primes and prove that the Diophantine inequality $|p_1^c+p_2^c+p_3^c+p_4^c+p_5^c+p_6^c-N|<{\log }^{-1}N$ is solvable in primes $p_1,p_2,p_3,p_4,p_5,p_6$ for sufficiently large real number $N$.

21 pages