The Most Controversial Proof in Mathematics Ever

Mathematics rests on a promise: if a proof is correct, a trained mathematician should be able to check it. Shinichi Mochizuki's claimed proof of the ABC conjecture has tested that promise, dividing proponents of his theory from prominent critics. This article explains the conjecture, why it matters, and the disagreement about whether the proposed argument establishes it.

A Proof Divided by Geography

In 2012, Shinichi Mochizuki, a mathematician at the Research Institute for Mathematical Sciences (RIMS) in Kyoto, released four papers on a new framework called inter-universal Teichmüller theory, or IUT. He claimed it proves the ABC conjecture. Mochizuki had earned a Princeton PhD at 23 and was respected for earlier work in anabelian geometry. The IUT papers were submitted in August 2012 and published in 2021 in a RIMS journal, where they fill roughly 720 pages of a single issue ^[1].

The papers lean on a long chain of earlier constructions and a large amount of new vocabulary, so few outsiders managed to read them closely. Support for the proof is now mostly limited to a small circle of researchers, which one commentator places in Kyoto and Nottingham ^[2].

The Scholze–Stix Objection

In March 2018, Peter Scholze and Jakob Stix spent a week in Kyoto in discussions with Mochizuki and Yuichiro Hoshi, focused largely on Corollary 3.12 of the third IUT paper ^[3]. They concluded that there is no proof, and that the problem is so severe that small modifications would not rescue the strategy ^[4]. Quanta Magazine reported Stix describing a serious, unfixable gap, while Mochizuki attributed the criticism to fundamental misunderstandings of his work ^[5].

Later, Kirti Joshi proposed his own arithmetic Teichmüller framework aimed at proving ABC ^[6]. Joshi has said his stance on whether Mochizuki proved ABC is neutral ^[7], so it is a simplification to say he merely repaired Mochizuki's argument. A prize of about one million dollars was also announced for a paper demonstrating a flaw in the claimed proof ^[8]. The disagreement remains unresolved. Readers may also wish to see how mathematicians think about other famous long-running problems in this survey of unsolved problems that were eventually solved.

What the ABC Conjecture Says

The conjecture concerns coprime positive integers with a + b = c. The radical of abc, written rad(abc), is the product of the distinct primes dividing abc. Take 1024 + 81 = 1105. Here 1024 = 2¹⁰, 81 = 3⁴ and 1105 = 5 × 13 × 17, so the radical is 2 × 3 × 5 × 13 × 17 = 6630.

The conjecture says that for every ε > 0, there is a constant C(ε) such that c is at most C(ε) times a power of rad(abc), with exponent 1 + ε, for all such triples ^[9]. Equivalently, that power of the radical exceeds c for all but finitely many triples. The exceptions, where c is unusually large compared with the radical, are called abc hits. The example above is not a hit, since 1105 is smaller than 6630. A classic hit is 2 + 3¹⁰ × 109 = 23⁵, whose radical is only 15042 ^[10]. For a related taste of number-theoretic puzzles that look easy but are not, see this article on unsolved prime number problems.

c ≤ C(ε) × rad(abc)^{1 + ε}

Why It Matters: Fermat's Last Theorem

If ABC is true, it implies Fermat's Last Theorem for sufficiently large exponents ^[9]. Suppose coprime positive integers satisfy xⁿ + yⁿ = zⁿ. The radical of xⁿyⁿzⁿ is at most xyz, and xyz < z³. ABC therefore bounds zⁿ by C(ε) times z^{3(1 + ε)}. The constant C(ε) must be retained; dropping it would make the argument too strong. For a fixed positive ε, this inequality bounds how large a possible counterexample's exponent can be ^[11]. Andrew Wiles proved the full theorem by different methods. For another story about the structure of proofs, see increasingly advanced proofs that 0.999… equals 1.

An Open Question About Verification

Mathematics has met disputed proofs before: the four color theorem relied on computers, and Perelman's work took years to digest. In those cases consensus eventually formed. Here, after more than a decade, the community still disagrees on whether a verifiable proof exists. Some mathematicians have wondered whether formal proof assistants such as Lean might one day help check arguments of this depth, though such ideas remain exploratory. Publication of the IUT papers has not ended the dispute over whether the proposed argument proves ABC ^[4].

Sources

1. Volume 57, No. 1/2 (2021) Special issue on Inter-universal Teichmüller Theory| Publications of the Research Institute for Mathematical Sciences
2. ABC is Still a Conjecture
3. Deep Disagreement in Mathematics
4. [PDF] Why abc is still a conjecture
5. Titans of Mathematics Clash Over Epic Proof of ABC Conjecture
6. Final Report on the Mochizuki-Scholze-Stix Controversy
7. The status of the Scholze-Stix Report and an analysis … – CDN
8. Million Dollar Prize for Scholze and Stix
9. abc Conjecture — from Wolfram MathWorld
10. Abc conjecture
11. ABC and FLT

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