PreprintarXiv licencearXiv
Note on the Exceptional Set in the ABC Conjecture
Abstract
Fix ε>0, let x>1 be a large real number and let rad(n)=∏p| np be the radical of an integer n≥1. A triple (a,b,c), with a+b=c and gcd(a,b,c)=1, such that c>(rad(abc))1+ε, is called exceptional triple. Recent works have proved that the cardinality #ℰ(x) of set ℰ of exceptional triples satisfies #ℰ(x)=O(x33/50). This note proves that the cardinality of the exceptional set ℰ(x) of triples (a,b,c) is an infinite set unconditionally for any ε<2/35.
Related
- A note on the proof of Barnette's conjecture
- A note on the p-adic inhomogeneous Littlewood conjecture
- A Note on the OpenAI Construction of Counterexamples to the Kaplansky Direct-Finiteness Conjecture
- A note on Kuperberg's quadrilateral conjecture
- A note on the noncommutative Hodge conjecture for graded matrix factorizations
BibTeX
@misc{carella2026note,
title = {{Note on the Exceptional Set in the ABC Conjecture}},
author = {N. A. Carella},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2608.16764},
eprint = {2608.16764},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.16764},
}
RIS
TY - UNPB
TI - Note on the Exceptional Set in the ABC Conjecture
AU - N. A. Carella
PY - 2026
DA - 2026/09/19
PB - arXiv
DO - 10.48550/arxiv.2608.16764
UR - https://arxiv.org/abs/2608.16764
AB - Fix ε>0, let x>1 be a large real number and let rad(n)=∏_p| np be the radical of an integer n≥1. A triple (a,b,c), with a+b=c and gcd(a,b,c)=1, such that c>(rad(abc))^1+ε, is called exceptional triple. Recent works have proved that the cardinality #ℰ(x) of set ℰ of exceptional triples satisfies #ℰ(x)=O(x^33/50). This note proves that the cardinality of the exceptional set ℰ(x) of triples (a,b,c) is an infinite set unconditionally for any ε<2/35.
ER -