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

Abstract

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