On the difference between perfect powers and integral $S$-units

Abstract

Let $q_1, \ldots , q_t$ be distinct prime numbers. Let $a_1, \ldots , a_t$ be nonnegative integers. We establish effective lower bounds for $|z^d - q_1^{a_1} \ldots q_t^{a_t}|$ and for its greatest prime factor, which tend to infinity with $z^d$, where $z$ is a positive integer coprime with $q_1 \ldots q_t$ and $d \ge 2$ is an integer.