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Mathematicians Still Years from Verifying Nightmare 500-Page Proof

Shinichi Mochizuki's proof of the abc conjecture remains an enigma.

by Michael Byrne
Jul 30 2016, 2:30pm

Image: Paddy Mills/Creative Commons

In 2012, the mathematician Shinichi Mochizuki published four papers on his own website with little fanfare. Together, they totalled more than 500 pages—a hyperdense labyrinth of symbols invoking a brutal new mathematical framework known as inter-universal Teichmüller Theory. Via said framework, Mochizuki claimed to have solved the abc conjecture—a simple-seeming statement about prime numbers that has nonetheless eluded mathematicians since first being proposed in 1985. The catch? No other mathematician could understand the thing.

Mochizuki had unleashed an impenetrable mathematical hellscape. Understanding it is not simply a matter of being a really great number theorist, one has to become an expert in an entirely new mathematical field as defined in Mochizuki's papers. The only preparation for verifying the new proof is to understand inter-universal Teichmüller Theory, which no one really does. Thus, a paradox is raised: verifying the proof means fully committing to it for years even though it might be wrong.

In a verification progress report published in December of 2014, Mochizuki bemoaned: "With the exception of the handful of researchers already involved in the verification activities concerning IUTeich (inter-universal Teichmüller Theory) discussed in the present report, every researcher in arithmetic geometry throughout the world is a complete novice with respect to the mathematics surrounding IUTeich, and hence, in particular, is simply not qualified to issue a definitive (i.e., mathematically meaningful) judgment concerning the validity of IUTeich on the basis of a 'deep understanding' arising from his/her previous research achievements."

Mathematicians have nonetheless taken up the challenge. At a workshop last week at the University of Kyoto, the normally reclusive Mochizuki presented his work in-person to a gathering of believers, several of whom noted significant progress in untangling the proof. UC San Diego number theorist Kiran Kedlaya at least sees a light at the end of the tunnel, however distant. He told Nature News that we can expect a verification no less than three years from now, but that mathematicians have begun to uncover a general strategy unlying Mochizuki's proof and have isolated passages that seem particularly key.

Moreover, there are now 10 mathematicians committed to the cause, up from three at the time of a workshop last December at Oxford.

Skepticism remains, however. "The constructions are generally clear, and many of the arguments could be followed to some extent, but the overarching strategy remains totally elusive for me," Yale mathematician Vesselin Dimitrov told Nature. "Add to this the heavy, unprecedentedly indigestible notation: these papers are unlike anything that has ever appeared in the mathematical literature." (Dimitrov, who is a recurring figure in the abc conjecture saga, had previously claimed to have found a contradiction in the proof, but Mochizuki was able to explain it away.)

In comments posted to Facebook (via Not Even Wrong), workshop organizer Ivan Fesenko, a mathematician at the University of Nottingham, noted that participants worked hard and kept a friendly atmosphere. At the end, they were "tired but happy."

Fesenko also lightly chided IUTeich skeptics (of which there are quite a few, including "Mathbabe" Cathy O'Neil), writing, "It is true that people who take relatively negative or sceptical positions towards IUT are often, if not always, the least interested or most reluctant to ask detailed questions. Ample opportunities to ask such questions were available during the workshop. Those who intensively used them visibly progressed in their understanding of IUT."

Proving the abc conjecture may prove to be worth the effort. A (very gnarly) paper by Dimitrov earlier this year showed how a reduction of Mochizuki's proof, if it is eventually verified, should offer solutions to a large number of other mathematical problems, including a new proof of Fermat's last theorem.

As for what the abc conjecture actually is, here's what I wrote in 2015:

Take three (whole) numbers—a, b, and c—of the form a + b = c. These three numbers are what's known as coprime. This just means that they don't share any divisors (except for 1, but every number is divisible by 1). If one of the numbers is divisible by, say, 13, none of the others are divisible by 13. It's an interesting relationship, like three people with no shared ancestors all the way back to the very first human/number (which is 1).

What we need to prove is that for some other, fourth number d, there will be a finite (countable) number of abc triplet sets (as, bs, and cs fitting the above equation) such that c is greater than d raised to some positive power that is not 1 (a "perfect power"). Finally, this extra number d isn't just anything; it must be the product of the prime factors of a, b, and c. That's it. Somehow, these three numbers that aren't linked by common factors (coprime) are, after all, linked by their factors. It's weird.

You could even say that the conjecture is kind of elegant: three seemingly perfect mathematical strangers are related in the end. A simple statement, but with a proof that turns out to be anything but.