On the exceptional set in the $abc$ conjecture

TL;DR

This paper improves bounds on the number of coprime triples satisfying a weakened form of the abc conjecture, using optimized methods to show the count grows slower than previously established.

Contribution

It introduces an enhanced bound on the count of triples related to the abc conjecture using refined techniques from recent research.

Findings

Number of triples with $c \,\leqslant\, X$ is $O(X^{56/85+\varepsilon})$

Improves previous bound of $O(X^{33/50})$

Demonstrates the effectiveness of optimized methods in this context.

Abstract

The $abc$ conjecture states that there are only finitely many triples of coprime positive integers $(a,b,c)$ such that $a+b=c$ and $\operatorname{rad}(abc) < c^{1-\epsilon}$ for any $\epsilon > 0$. Using the optimized methods in a recent work of Browning, Lichtman and Ter\"av\"ainen, we showed that the number of those triples with $c \leqslant X$ is $O\left(X^{56/85+\varepsilon}\right)$ for any $\varepsilon > 0$, where $\frac{56}{85} \approx 0.658824$. This constitutes an improvement of the previous bound $O\left(X^{33/50}\right)$.