Rational and algebraic approximations of algebraic numbers and their application

摘要

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 D0 is not a square,and ε=x0 +y0 D denotes the fundamental solution of x2-Dy2=-1,then x2+1=Dy4 is solvable if and only if y0=A2 where A is an integer.Moreover,if ≥64,then x2+1=Dy4 has at most one positive integral solution (x,y).