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

TL;DR

This paper provides effective lower bounds for the difference between perfect powers and products of prime powers, showing these bounds grow with the size of the perfect power.

Contribution

It establishes explicit lower bounds for the difference and greatest prime factor of perfect powers minus products of prime powers, extending understanding of their arithmetic properties.

Findings

Lower bounds tend to infinity with the perfect power.

Bounds are effective and explicit.

Results apply to coprime integers and powers with exponent at least 2.

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.