On the exceptional set in the $abc$ conjecture

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 $\operatorname{rad}(abc) &lt; c<sup>{1-\epsilon}$ for any $\epsilon &gt; 0$. 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 $O\left(X<sup>{56/85+\varepsilon}\right)$ for any $\varepsilon &gt; 0$, where $\frac{56}{85} \approx 0.658824$. This constitutes an improvement of the previous bound $O\left(X<sup>{33/50}\right)$.