ABC conjecture and its exceptional-set problem

The exceptional-set problem asks whether triples violating the ABC inequality can occur infinitely often. A lone-author preprint dated August 17, 2026 claims that, for fixed small $\varepsilon>0$, infinitely many coprime triples satisfy $a+b=c$ and $c>\operatorname{rad}(abc)^{1+\varepsilon}$, which would contradict the standard ABC conjecture.

Known results

• Browning, Lichtman, and Teräväinen, 2024: $N_\lambda(X)=O(X^{33/50})$ for fixed $\lambda\in(0,1.001)$, improving the earlier $O(X^{2/3})$ scale near $\lambda=1$.
• 2025 work: $N_\lambda(X)\ll X^{(23\lambda+3)/40+\varepsilon}$ for $\lambda\in(0,2)$, giving $N_1(X)\ll X^{0.65+\varepsilon}$.
• Mochizuki’s claimed proof of the ABC conjecture remains unaccepted; Scholze and Stix reported a serious gap.

August 2026 claimed infinite exceptional set

N. A. Carella’s preprint states that the counting function of such exceptional triples tends to infinity. This is a claim, not a verified or refereed result; no corroborating proof, objection report, withdrawal, or retraction was found.

Current status (as of August 2026): The ABC conjecture and the claimed infinite-exceptional-set result remain unverified; only substantial upper bounds on the number of potential exceptions are established.