Simplifying method for algebraic approximation of certain algebraic numbers

A new simple method for approximating certain algebraic numbers is developed. By applying this method, an effective upper bound is derived for the integral solutions of the quartic Thue equation with two parameters \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$tx^4 - 4sx^3 y - 6tx^2 y^2 + 4sxy^3 + ty^4 = N$\end{document}, where s > 32t3. As an application, Ljunggren’s equation is solved in an elementary way.