Sums of two ${s}$-units via frey-hellegouarch curves

TL;DR

This paper introduces a novel method leveraging modularity of Galois representations to find perfect powers expressed as sums of two rational S-units, successfully applied to specific small prime sets.

Contribution

It presents a new approach that avoids lower bounds for linear forms in logarithms, enabling explicit solutions for certain small prime sets S.

Findings

Successfully applied to S = {2, 3} and S = {3, 5, 7}

Develops a method based on modularity of Galois representations

Advances the computational techniques for sums of two S-units

Abstract

In this paper, we develop a new method for finding all perfect powers which can be expressed as the sum of two rational S-units, where S is a finite set of primes. Our approach is based upon the modularity of Galois representations and, for the most part, does not require lower bounds for linear forms in logarithms. Its main virtue is that it enables to carry out such a program explicitly, at least for certain small sets of primes S; we do so for S = {2, 3} and S = {3, 5, 7}.