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arXiv · 2507.02885

On the exceptional set in the $abc$ conjecture

Runbo Li
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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 $O\left(X^{33/50}\right)$.

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Runbo Li. 2025-06-19. On the exceptional set in the $abc$ conjecture. https://arxiv.org/abs/2507.02885

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