The A-B-C's of the abc Conjecture

Scatter plot of Pythagorean triples. Dearjean13, CC BY-SA 4.0 https://creativecommons.org/licenses/by-sa/4.0, via Wikimedia Commons

In 1994, the mathematics was turned on its head by Andrew Wiles’s proof of Fermat’s last theorem, a deceptively simply math problem asking if there are any integer (that is, positive whole number) triples $a, b,$ and $c$ that satisfy $a^n + b^n = c^n$ when $n > 2$. When $n = 1$, this is just boring addition. When $n = 2$, this is the Pythagorean theorem, with Pythagorean triples as the solution. When $n = 3, 4, 5…$ there are always the “trivial” solutions $(1, 0, 1)$, $(0, 1, 1)$, or $(0, 0, 0)$, but there didn’t seem to be any others. Solving Fermat’s last theorem (or just “FLT” for short) would be either proving the existence of a nontrivial solution, or proving no such solution exists.

It took over 350 years for Andrew Wiles to prove that there are no such nontrivial solutions, by way of a titanic treatise on elliptic curves and modular forms. Wiles’s paper opened the door to a fascinating and deep connection between number theory and Fourier analysis, part of a larger project known as the Langlands program.

But before Wiles published his proof, most mathematicians would have guessed that the most likely proof of FLT would come by way of a proof of another conjecture, called the $abc$ conjecture.

To explain the \(abc\) conjecture, we first have to set up some terminology. Given an integer \(n\), the radical of $n$, denoted $\text{rad}(n)$, is the product of each of the distinct prime factors of $n$. $12 = 3 \cdot 2^2$, so $\text{rad}(12) = 3 \cdot 2 = 6$. $80 = 5 \cdot 2^4$, so $\text{rad}(80) = 5 \cdot 2 = 10$, and so on. For any prime $p$ (or a positive power of $p$, like $p^2$ or $p^3$) we have $\text{rad}(p) = \text{rad}(p^2) = \text{rad}(p^3) = p$, and so on.

The $abc$ conjecture essentially says that if we have three whole numbers $a$, $b$, and $c$ which share no prime factors, then if $a + b = c$, then $c < \text{rad}(a\cdot b \cdot c)$ “almost always”. More specifically, if we choose a positive real number $\epsilon$, then there are only finitely many whole number triples $(a, b, c)$ that share no prime factors and also satisfy $a + b = c$ and $c > \text{rad}(a \cdot b \cdot c)^{1 + \epsilon}$. We typically use $\epsilon$ to denote quantities we think of as being small, so this means there are only finitely many of these triples where $c > \text{rad}(a \cdot b \cdot c)^{1.1}$ (here $\epsilon = 0.1$) or $c > \text{rad}(a \cdot b \cdot c)^{1.005}$ (here $\epsilon = 0.005$) and so on. Obviously, we expect the conjecture to still hold even if $\epsilon = 5$ or $\epsilon = 200$ etc. If the $abc$ conjecture is true, there will only ever be finitely many $a, b, c$ for any value of $\epsilon$ we could choose.

There’s an almost one-line proof that \(abc\) implies FLT, and there are numerous other other conjectures which would be instantly proven if \(abc\) were settled. Among reasons to believe the \(abc\) conjecture is true is that a similar statement for polynomials, called the Mason-Stothers theorem, has already been proven. (As has its corollary, Fermat’s last theorem for polynomials.) Many theorems about integers are true for polynomials and vice versa. After all, polynomials with rational or real coefficients both form a Euclidean domain, so they behave strikingly like integers: we can add, subtract, multiply, and factor polynomials. There is even a notion of “prime” polynomials!

If none of that made any sense, don’t worry. This article isn’t actually about the $abc$ conjecture as it is the controversy around a purported proof of $abc$.

Until 2012, it seemed the $abc$ conjecture resisted any attempts at proof. Then, in 2012, Shinichi Mochizuki published a series of four papers totalling about 500 pages claiming to solve the $abc$ conjecture. His papers develop a new field of math called inter-universal Teichmüller theory (abbreviated IUT or IUTT), which is closely related to the more well-established field of anabelian geometry.

In mathematics, it’s very common publish pre-prints (usually through arXiv.org). From there, the paper is subjected to peer review. Determining whether a proof is correct or flawed is sometimes very difficult even for experts especially when new tools or techniques are used, so authors typically spend the period between pre-print publication and journal publication answering questions and addressing concerns from reviewers and the wider mathematical community. Inevitably, mistakes – some fixable, some not – are found, and usually consensus on the correctness of a piece of mathematics is reached between several months to several years, depending on the difficulty and volume of the work.

So, what separates the wheat from the chaff? A mathematical proof must be correct, complete, and convincing. Correct means that the proof does what it claims to do: it constitutes a sequence of mathematically valid inferences from some starting point, called the axioms, to the end goal, called the result. Mathematicians use the word “proof” rather than “evidence” because a proof is supposed to provide absolute certainty that the axioms necessarily imply the result. Whether a proof is correct or not is objective: it either constitutes a valid sequence of valid inferences, or it doesn’t. (Compare this to the natural sciences, where theories are not proven but rather supported or not supported by evidence. Scientists are always open to the possibility that new evidence may contradict an established theory; in mathematics, if a new result contradicts an existing one, something has gone wrong.)

But correctness alone is not sufficient. A proof must also be complete: it cannot contain large gaps between the various steps in the proof. It’s common for the sake of clarity and brevity to skip small or uninteresting steps or calculations if a trained mathematician could reasonably notice and fill in those missing parts themselves. However, sometimes an author misjudges how large a gap is acceptable, or accidentally makes a jump in their reasoning that they have not justified. In this case, the proof is not complete, and peer reviewers will flag these gaps and raise their concerns with the author.

Finally, a given proof must be convincing. For a proof to be accepted and published, it must convince the peer reviewers that it is in fact correct and complete. This typically means that the author will organize their paper in a clear and easily digestible manner, provide commentary wherever necessary, and so on. It doesn’t matter if a proof is complete and correct if no one understands it well enough to be convinced!

Unfortunately, Mochizuki’s 2012 series of papers is utterly inscrutable even to seasoned mathematicians. Most peer reviewers simply gave up vetting the proof, especially when the concerns they raised to Mochizuki were either ignored or met with an insistence that they had not tried hard enough to understand his work. Eventually, Mochizuki’s circle settled on the consensus that his proof was correct; the rest of the world was not convinced.

Normally, this wouldn’t be a problem: lots of proofs not accepted by the wider mathematical community are published on arxiv and elsewhere on the internet every day. No one cares if some random person claims to have disproven the Pythagorean theorem and convinces his friends. But Mochizuki is not a random person: Mochizuki is a world-famous specialist in number theory and arithmetic geometry, and holds a position at Kyoto University, and chairs the journal Publications of the Research Institute for Mathematical Sciences (PRIMS). In summary: Mochizuki is not a nobody.

But his detractors aren’t nobodies either. The two most prominent ones are Peter Scholze and Jakob Stix, two specialists in anabelian geometry – a field closely related to Mochizuki’s work. Scholze is considered one of the world’s leading mathematicians: he received a Fields medal in August 2018, which is roughly analogous to a “Nobel Prize in mathematics” and is widely considered the highest distinction a mathematician can attain. Stix is no slouch either: he’s a laureate of the Institute for Advanced Study, one of the most prestigious research institutions in the world.

In August 2018, Scholze and Stix published a paper asserting that Mochizuki’s work was not only severely flawed, but unfixable. Scholze and Stix’s objections were published after the pair traveled to Kyoto in March of 2018 to discuss the proof with Mochizuki. By the end of the trip, Scholze and Stix had pinpointed a perceived flaw, and Mochizuki’s rebuttal left them unconvinced. So, Scholze and Stix collected their arguments and published them, and Mochizuki did the same for his responses, both hosted on Mochizuki’s website. (Please go to Mochizuki’s website.)

Several of Mochizuki’s comments in these rebuttals are a bit unusual.

Responding to one of the Scholze and Stix’s remarks on his work, Mochizuki wrote

“I can only say that it is a very challenging task to document the depth of my astonishment when I first read this Remark! This Remark may be described as a breath-takingly (melo?)dramatic self-declaration, on the part of SS, of their profound ignorance of the elementary theory of heights, at the advanced undergraduate/beginning graduate level.”

The controversy became a popular spectacle at this point with an excellent Quanta article covering the drama, at which point the controversy attracted much more attention.

By 2021, Mochizuki’s journal had published his results, which is typically a major milestone for the acceptance of mathematics. Despite recusing himself from the review process, the wider mathematical wasn’t moved by the publication of his work considering his close ties with the rest of the editorial board. Consensus had already settled that Mochizuki’s proof was not convincing and likely not even correct. By this point, anyone interested in seeing Scholze and Stix’s response through Mochizuki’s webpage had to click through an intermediary webpage containing a rebuttal by Mochizuki. Mochizuki also published a further defense of his work, which has grown from 65 pages in 2021 to 167 in March 2024.

It is a masterpiece.

It begins with a scathing criticism of Scholze and Stix, who he calls the “redundant copies school” or just “RCS”. Peter Woit of Columbia University summarized Mochizuki’s response as follows:

Essentially the claim Mochizuki is making in these first two sections is that the most accomplished and talented young mathematician in his field is an ignorant incompetent, and that everyone Mochizuki has consulted about this agrees with him. It’s hard to imagine a more effective way to destroy one’s own credibility and to convince people not to bother…

There are a myriad of fascinating digressions in Mochizuki’s writings; I’ll present a handful of selections. One section, 1.9.1, is an anonymized anecdote of students submitting papers for review:

This student (St1) showed his paper to a prominent senior researcher (Pf1) at a university in country (Ct2) in a certain area of number theory. The education and career of this researcher (Pf1) was conducted entirely at universities in countries (Ct2), (Ct3), and (Ct4). This researcher (Pf1) informed (St1) of his very positive evaluation of the originality of the results obtained in the paper by (St1).

This is ostensibly part of a larger critique of favoritism in academia, and continues on until there are eleven countries, seven professors, and two students:

Finally, it should be mentioned that each official language of each of these countries (Ct1), (Ct2), (Ct3), (Ct4), (Ct5), (Ct6), (Ct7) belongs to the European branch of the Indo-European family of languages, and that at least six of the ten pairs of countries in the list (Ct2), (Ct3), (Ct4), (Ct6), (Ct7) share a common official language [i.e., with the other country in the pair].

I have almost no clue what it means, or why the languages and nationalities of each of the mathematicians involved are relevant, but this is a common thread. In fact, it grows out into a bit of political philosophy: from section 1.10,

From a historical point of view, various forms of institutional and conceptual infrastructure — such as the notions of
· a modern judiciary system;
· universal, inalienable human rights;
· the rule of law;
· due process of law; and
· burden of proof
— were gradually developed, precisely with the goal of averting the outbreak of the sort of socio-political dynamics that were viewed as detrimental to society

Example 1.5.1 is named “Irrationality, impartiality, and the Voodoo Hypothesis”:

Once the Voodoo Hypothesis has been invoked, mathematicians who have a mathematically accurate, rigorous understanding of the proof in question are then often portrayed as being nothing more than **mindlessly obedient zombies**, i.e., who are acting solely or essentially under the influence of the “voodoo” applied in the proof.

I am certain that one could skip to any random page in this work and find something interesting. In the words of Columbia professor Peter Woit, “I don’t think it will convince anyone Scholze is wrong about the flaw in Mochizuki’s proof.”

There is one part of Mochizuki’s response where he characterizes Scholze and Stix’s discussion of IUT as a strawman of Mochizuki’s work, generating drama:

Such lurid dramas then spawn further grotesquely distorted mass media reports and comments on the English-language internet…

Guilty as charged.

By this point, everyone outside of Mochizuki’s circle is convinced that Mochizuki’s proof is incorrect, there is little of interest in IUT, and it is fruitless to continue engaging. However, one mathematician by the name of Kirti Joshi has fallen somewhere between the two camps: he claims that Mochizuki’s theory is capable of proving $abc$, but needs significant clarification. He also claims that Scholze and Stix have misunderstood some of Mochizuki’s work. Scholze has already responded, and claims that Joshi’s work cleaning up IUT still doesn’t resolve fundamental problems with the theory. Mochizuki, as expected, had an even less diplomatic response to Joshi’s work:

although Joshi, in this series of preprints, makes references to and often uses certain portions of the terminology and notation of inter-universal Teichmüller theory, it is conspicuously obvious to any reader of these preprints who is equipped with a solid, rigorous understanding of the actual mathematical content of inter-universal Teichmüller theory that the author of this series of preprints is profoundly ignorant _of the actual mathematical content of inter-universal Teichmüller theory*, and, in particular, that this series of preprints does not contain, at least from the point of view of the mathematics surrounding inter-universal Teichmüller theory, any meaningful mathematical content whatsoever.

Eleven more pages follow the above paragraph, with a similarly caustic tone:

I could not help but be re-minded of the so-called “hallucinations” produced by artificial intelligence algorithms, such as ChatGPT, i.e., which are synthesized precisely by means of various mechanically searched contextual concatenations that are entirely devoid of any genuine “human” understanding of the actual content of the text involved.

One comment, on a theorem Joshi has numbered 9.11 (which is a common numbering scheme in mathematical texts), where Mochizuki seemingly implies Joshi is making a 9/11 joke:

[where we note that it is not clear whether or not the number “9.11…” assigned by the author to these key results in [CnstIII] was purely coincidental or a consequence of some sort of sense of rhetoric or humor that lies beyond my understanding]

As for Scholze, he seems unconvinced despite Joshi’s discussions that that IUT is even a viable method to prove $abc$ (although Joshi says talks were ongoing as of late June 2024).

So, where does this leave us? Only Mochizuki and a handful of close colleagues accept his work, and his reputation has incurred enough self-inflicted wounds that he is likely no longer worth the time of day to mathematicians outside his circle. Joshi is one of the few outside of that circle who claims Mochizuki’s work is salvageable, but so far seems alone in his position. Scholze’s attention is likely to be spent on more promising work, like a recent solution to the geometric Langlands conjecture, so I doubt we’ll get an update anytime soon. To conclude: as Peter Woit said in 2021, $abc$ is still a conjecture.