Counting solvable $S$-unit equations

TL;DR

This paper establishes upper bounds on the number of prime sets below a certain limit that allow solutions to specific two-variable S-unit equations, advancing understanding of their solvability.

Contribution

It provides new upper bounds on the count of prime sets for which certain S-unit equations are solvable, a novel result in the study of these equations.

Findings

Derived explicit upper bounds for prime sets solving S-unit equations

Extended previous results on the solvability of two-variable S-unit equations

Improved understanding of the distribution of prime sets related to these equations

Abstract

We obtain upper bounds on the number of finite sets $\mathcal S$ of primes below a given bound for which various $2$ variable $\mathcal S$-unit equations have a solution.