On the difference between squares and integral $S$-units

Abstract

Let $q_1, \ldots , q_t$ be distinct prime numbers. Let $a_1, \ldots , a_t$ be nonnegative integers and $x$ a positive integer. We establish an effective lower bound for the greatest prime divisor of $|x^2 - q_1^{a_1} \ldots q_t^{a_t}|$, which tends to infinity with the maximum of $x$, $a_1, \ldots , a_t$.