ABC conjecture and its exceptional-set problem

The ABC conjecture asserts that, for every fixed positive margin above $1$, only finitely many primitive solutions to $a+b=c$ violate the corresponding radical bound. Shinichi Mochizuki claimed a proof in 2012, but its acceptance remains disputed.

Known results

• Mochizuki, 2012: claimed a proof via inter-universal Teichmüller theory; it has not gained mathematical consensus.
• Scholze and Stix, 2018: reported a serious gap near Corollary 3.12; the conjecture was regarded as open.
• Browning, Lichtman, and Teräväinen, 2024–2025: improved exceptional-set counts to $N_\lambda(X)\ll X^{33/50}$, and subsequent work claims $N_\lambda(X)\ll X^{56/85+\varepsilon}$; these bounds do not imply finiteness.

August 17, 2026 claimed disproof

An arXiv note claims that, for every sufficiently small fixed $\varepsilon>0$, infinitely many primitive triples satisfy $c>\operatorname{rad}(abc)^{1+\varepsilon}$, allegedly constructing them from smooth integers in short intervals. If correct, this directly refutes the stated exceptional-set assertion; the claim is unverified.

Current status (as of August 2026): Mochizuki’s proof remains unaccepted, earlier bounds are partial, and the August 17, 2026 claimed disproof is not independently verified.