On a Diophantine inequality involving prime powers

Suppose that N is a sufficiently large real number. In this paper it is proved that if 1 < c < 108/53, c ≠ 2, then the Diophantine inequality \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${| p^c_1 + p^c_2+ p^c_3 +p^c_4+ p^c_5 - N| < \log^{-1}N}$$\end{document} is solvable in primes p1, p2, p3, p4, p5. This result constitutes an improvement upon that of Zhai and Cao for the range 1 < c < 81/40, c ≠ 2.