Exceptional set for sums of unlike powers of primes (II)

Let N be a sufficiently large integer. In this paper, it is proved that, with at most O(N7/18+ε)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$O(N^{7/18+\varepsilon })$$\end{document} exceptions, all even positive integers up to N can be represented in the form p12+p22+p33+p43+p54+p64\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p_1^2+p_2^2+p_3^3+p_4^3+p_5^4+p_6^4$$\end{document}, where p1,p2,p3,p4,p5,p6\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p_1,p_2,p_3,p_4,p_5,p_6$$\end{document} are prime numbers, which constitutes an improvement over some previous work.