Sums of Five Almost Equal Prime Squares

Let Pi, 1 ≤ i ≤ 5, be prime numbers. It is proved that every integer N that satisfies N ≡ 5(mod24) can be written as \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ N = p^{2}_{1} + p^{2}_{2} + p^{2}_{3} + p^{2}_{4} + p^{2}_{5} ,{\text{where}}{\left| {{\sqrt N }5 - p_{i} } \right|} \leqslant N^{{\frac{1} {2} - \frac{{19}} {{850}} + \in }} $$\end{document}.