engleski

On some applications of a theorem of Worley

The following result of Worley and Dujella extends classical results of Legendere and Fatou concerning Diophantine approximations of the form |alpha - a/b| < 1/2b^2
and |alpha - a/b| < 1/b^2: Let alpha be a real number and a and b coprime nonzero integers, satisfying the inequality |alpha - a/b| < M/b^2 ; where M is a positive real number. Then (a, b) = (rp_{;n+1}; +- up_n, rq_{;n+1}; +- uq_n), for some n  >= -1 and nonnegative integers r and u such that ru < 2M ; where p_n=q_n denotes the n-th convergent of alpha. This result is very useful for solving various types of problems that can be reduced to solving a parametric Pellian equation or a system of the parametric Pellian equations with bounded right hand side. In this talk we will present a few results obtained in this way.

diophantine approximations; continued fractions; Thue equations

nije evidentirano