Smooth solutions to the abc equation: the xyz Conjecture

TL;DR

This paper investigates the distribution of solutions to the ABC equation with constraints on prime factors, showing finiteness under the abc Conjecture for certain smoothness levels and infinite solutions under GRH for larger smoothness.

Contribution

It provides conditional results linking the abc Conjecture and GRH to the existence and finiteness of smooth solutions to the ABC equation.

Findings

Finiteness of solutions if smoothness p<1 under abc Conjecture.

Infinitely many solutions if p>8 assuming GRH.

Sketches of proofs connecting number theory conjectures to solution existence.

Abstract

This paper studies integer solutions to the ABC equation A+B+C=0 in which none of A, B, C has a large prime factor. Set H(A,B, C)= max(|A|,|B|,|C|) and set the smoothness S(A, B, C) to be the largest prime factor of ABC. We consider primitive solutions (gcd(A, B, C)=1) having smoothness no larger than a fixed power p of log H. Assuming the abc Conjecture we show that there are finitely many solutions if p<1. We discuss a conditional result, showing that the Generalized Riemann Hypothesis (GRH) implies there are infinitely many primitive solutions when p>8. We sketch some details of the proof of the latter result.