Geometric progressions in the sets of values of rational functions

Abstract

Let $a,Q\in Q$ be given and consider the set $G(a,Q)=\{a{Q}^i:i\in N\}$ of terms of geometric progression with 0th term equal to $a$ and the quotient $Q$. Let $f\in Q(x,y)$ and $V_f$ be the set of finite values of $f$. We consider the problem of existence of $a,Q\in Q$ such that $G(a,Q)\subset V_f$. In the first part of the paper we describe certain classes of rational functions for which our problem has a positive solution. In the second, experimental, part of the paper we study the stated problem for the rational function $f(x,y)=(y^2-x^3)/x$. We relate the problem to the existence of rational points on certain elliptic curves and present interesting numerical observations which allow us to state several questions and conjectures.