{"paper":{"title":"On the exceptional set in the $abc$ conjecture","license":"http://creativecommons.org/licenses/by-sa/4.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Runbo Li","submitted_at":"2025-06-19T06:17:20Z","abstract_excerpt":"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 $O\\left(X^{33/50}\\right)$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2507.02885","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2507.02885/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}