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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2507.02885 |
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| _version_ | 1866911037977001984 |
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| author | Li, Runbo |
| author_facet | Li, Runbo |
| contents | The $abc$ conjecture states that there are only finitely many triples of coprime positive integers $(a,b,c)$ such that $a+b=c$ and $\operatorname{rad}(abc) < c^{1-ε}$ for any $ε> 0$. Using the optimized methods in a recent work of Browning, Lichtman and Teräväinen, 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 $O\left(X^{33/50}\right)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_02885 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the exceptional set in the $abc$ conjecture Li, Runbo Number Theory The $abc$ conjecture states that there are only finitely many triples of coprime positive integers $(a,b,c)$ such that $a+b=c$ and $\operatorname{rad}(abc) < c^{1-ε}$ for any $ε> 0$. Using the optimized methods in a recent work of Browning, Lichtman and Teräväinen, 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 $O\left(X^{33/50}\right)$. |
| title | On the exceptional set in the $abc$ conjecture |
| topic | Number Theory |
| url | https://arxiv.org/abs/2507.02885 |