Finiteness properties of abc-equations c = a+b

TL;DR

This paper classifies finite sets of integer abc-equations based on their radicals, proving finiteness of classes using Mahler's 1933 theorem, advancing understanding of abc-equation properties.

Contribution

It introduces a classification scheme for abc-equations based on radicals and proves finiteness of classes using classical number theory results.

Findings

Finite classification of abc-equations by radical R(abc)

Proof of finiteness of equivalence classes

Application of Mahler's theorem to abc-equations

Abstract

We classify integer abc-equations c = a + b (to be defined), according to their radical R(abc) and prove that the resulting equivalence classes contain only a finite number of such equations. The proof depends on a 1933 theorem of Kurt Mahler.