The ABC conjecture is a deep conjecture relating the additive and multiplicative structure of integers. Though simply stated, it would imply many of the hardest theorems in number theory — including a relatively short proof of Fermat’s Last Theorem, the Mordell conjecture with effective bounds, and much more.
The statement
For a positive integer , let the radical be the product of its distinct prime factors. For example, .
The ABC conjecture (in one standard form):
For every , there are only finitely many triples of coprime positive integers with and
Intuitively: if are “additively simple” (they sum up) but “multiplicatively complicated” (they have many small prime factors), something unusual has to happen — and it can’t happen too often.
Why it matters
The ABC conjecture, if true, would imply a huge number of other results:
- Fermat’s Last Theorem (with effective bounds): a far shorter proof than Wiles’s, if ABC were a theorem.
- The Mordell conjecture with effective bounds.
- Catalan’s conjecture and generalizations.
- Results about Diophantine equations of many kinds.
- Bounds on prime gaps in certain arithmetic progressions.
ABC is often called “the most important conjecture in Diophantine equations.”
The Mochizuki controversy
In August 2012, the Japanese mathematician Shinichi Mochizuki of RIMS Kyoto posted four papers totaling about 600 pages, claiming to prove the ABC conjecture. His proof used a framework he had developed over the preceding decade, called Inter-universal Teichmüller theory (IUTT).
The mathematical community has struggled to verify the proof. IUTT is highly non-standard, introduces many new concepts, and has proved difficult for outside experts to evaluate. Several high-profile mathematicians — including Peter Scholze and Jakob Stix — have identified what they believe is a serious gap in a key step (Corollary 3.12 of IUTT III). Mochizuki has responded but the disagreement has not been resolved.
Japanese journals have published the papers, effectively endorsing them. Most of the international mathematical community remains unconvinced.
As of 2025, the status of ABC is: contested. It is not yet a theorem in the standard sense of a proof accepted by the community.
Cultural significance
The ABC controversy is the most prominent example in recent memory of a mathematical claim that the community cannot verify. It raises genuine questions about the sociology and pragmatics of mathematical knowledge: what does it mean for a theorem to be “proved” if only a small group of specialists can follow the proof?
These questions — about peer review, verification, and the role of communities in establishing truth — are likely to matter even more as proofs grow longer and more technical, and as machine-assisted proofs become common.
Frequently asked
Has ABC been proved?
Controversial. Shinichi Mochizuki in Japan published a claimed proof in 2012 using his own framework of 'Inter-universal Teichmüller theory' (IUTT). Most of the mathematical community has not accepted it. The debate remains unresolved as of 2025.
What would ABC imply if true?
A lot. It would give a nearly-immediate proof of Fermat's Last Theorem (with effective bounds), imply Mordell's conjecture with effective bounds, and resolve many Diophantine equations. It is one of the most powerful conjectures in number theory.