Insulators of ABC Solutions

Abstract

This note studies various ways of measuring the complexity of primitive solutions to the equation $A+B+C=0$ and conjectures relating them. We define the insulator $\mathcal{I}(A,B,C)$ of a primitive solution as the smallest positive integer $\mathcal{I}$ such that the primes dividing the product $ABC\cdot \mathcal{I}$ are exactly those below a given bound. We show that the strong XYZ Conjecture of Lagarias and Soundararajan implies there are only finitely many primitive solutions $(A,B,C)$ with a given insulator.

arXiv comment5 pages