Rational and algebraic approximations of algebraic numbers and their application

Effective rational and algebraic approximations of a large class of algebraic numbers are obtained by Thue-Siegel’s method. As an application of this result, it is proved that: if D>0 is not a square, and ε =x ₀ denotes the fundamental solution ofx ²?Dy ²=?1, thenx ²+1=Dy ⁴ is solvable if and only ify ₀=A ², where A is an integer. Moreover, if ε>64, thenx ²+1=Dy ⁴ has at most one positive integral solution (x, y).