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Easy as ABC? Not quite!

A brilliant mathematician named Shinichi Mochizuki claims to have proved the famous "abc conjecture" in number theory. That's great! There's just one problem: his proof is about 500 pages long, and almost nobody understands it, so mathematicians still can't tell if it's correct.

Luckily another mathematician named Go Yamashita has just written a summary of the proof. That's great! There's just one problem: it's 294 pages long, and it seems very hard to understand.

I'm no expert on number theory, so my opinion doesn't really matter. What's hard for me to understand may be easy for an expert!

But the most disturbing feature to me is that this new paper contains many theorems whose statements are over a page long... with the proof being just "Follows from the definitions".

Of course, every true theorem follows from the definitions. But the proof usually says how.

It's common to omit detailed proofs when one is summarizing someone else's work. But even a sketchy argument would help us understand what's going on.

This is part of a strange pattern surrounding Mochizuki's work. There was a conference in Oxford in 2015 aimed at helping expert number theorists understand it. Many of them found it frustrating. Brian Conrad wrote:

I don’t understand what caused the communication barrier that made it so difficult to answer questions in the final 2 days in a more illuminating manner. Certainly many of us had not read much in the papers before the meeting, but this does not explain the communication difficulties. Every time I would finally understand (as happened several times during the week) the intent of certain analogies or vague phrases that had previously mystified me (e.g., “dismantling scheme theory”), I still couldn’t see why those analogies and vague phrases were considered to be illuminating as written without being supplemented by more elaboration on the relevance to the context of the mathematical work.

At multiple times during the workshop we were shown lists of how many hours were invested by those who have already learned the theory and for how long person A has lectured on it to persons B and C. Such information shows admirable devotion and effort by those involved, but it is irrelevant to the evaluation and learning of mathematics. All of the arithmetic geometry experts in the audience have devoted countless hours to the study of difficult mathematical subjects, and I do not believe that any of us were ever guided or inspired by knowledge of hour-counts such as that. Nobody is convinced of the correctness of a proof by knowing how many hours have been devoted to explaining it to others; they are convinced by the force of ideas, not by the passage of time.

It's all very strange. Maybe Mochizuki is just a lot smarter than than us, and we're like dogs trying to learn calculus. Experts say he did a lot of brilliant work before his proof of the abc conjecture, so this is possible.

But, speaking as one dog to another, let me tell you what the abc conjecture says. It's about this equation:

a + b = c

Looks simple, right? Here a, b and c are positive integers that are relatively prime: they have no common factors except 1. If we let d be the product of the distinct prime factors of abc, the conjecture says that d is usually not much smaller than c.

More precisely, it says that if p > 1, there are only finitely many choices of relatively prime a,b,c with a + b = c and

d^p < c

It looks obscure when you first see it. It's famous because it has tons of consequences! It implies the Fermat–Catalan conjecture, the Thue–Siegel–Roth theorem, the Mordell conjecture, Vojta's conjecture (in dimension 1), the Erdős–Woods conjecture (except perhaps for a finitely many counterexamples)... blah blah blah... etcetera etcetera.

Let me just tell you the Fermat–Catalan conjecture, to give you a taste of this stuff. In fact I'll just tell you one special case of that conjecture: there are at most finitely many solutions of

x^3 + y^4 = z^7

where x,y,z are relatively prime positive integers. The numbers 3,4,7 aren't very special - they could be lots of other things. But the Fermat–Catalan conjecture has some fine print in it that rules out certain choices of these exponents. In fact, if we rule out those exponents and also certain silly choices of x,y,z, it says there are only finitely many solutions even if we let the exponents vary! Here's a complete list of known solutions:

1^m + 2^3 = 3^2
2^5 + 7^2 = 3^4
13^2 + 7^3 = 2^9
2^7 + 17^3 = 71^2
3^5 + 11^4 = 122^2
33^8 + 1549034^2 = 15613^3
1414^3 + 2213459^2 = 65^7
9262^3 + 15312283^2 = 113^7
17^7 + 76271^3 = 21063928^2
43^8 + 96222^3 = 30042907^2

The first one is weird because m can be anything: we need some fine print to say this doesn't count as infinitely many solutions.

It's a long way from here to the very first paragraph in the summary at the start of Yamashita's paper:

By combining a relative anabelian result (relative Grothendieck Conjecture over sub-p-adic felds (Theorem B.1)) and "hidden endomorphism" diagram (EllCusp) (resp. "hidden endomorphism" diagram (BelyiCusp)), we show absolute anabelian results: the elliptic cuspidalisation (Theorem 3.7) (resp. Belyi cuspidalisation (Theorem 3.8)). By using Belyi cuspidalisations, we obtain an absolute mono-anabelian reconstruction of the NF-portion of the base field and the function field (resp. the base field) of hyperbolic curves of strictly Belyi type over sub-p-adic fields (Theorem 3.17) (resp. over mixed characteristic local fields (Corollary 3.19)). This gives us the philosophy of arithmetical holomorphicity and mono-analyticity (Section 3.5), and the theory of Kummer isomorphism from Frobenius-like objects to etale-like objects (cf. Remark 3.19.2).

And it's a long way from this – which still sounds sorta like stuff I hear
mathematicians say – to the scary theorems that crawl out of their caves around page 200!

Check out Yamashita's paper and see what I mean:

You can read Brian Conrad's story of the Oxford conference here:

You can learn more about the abc conjecture here:

And you can learn more about Mochizuki here:

He is the leader of and the main contributor to one of major parts of modern number theory: anabelian geometry. His contributions include his famous solution of the Grothendieck conjecture in anabelian geometry about hyperbolic curves over number fields. He initiated and developed several other fundamental developments: absolute anabelian geometry, mono-anabelian geometry, and combinatorial anabelian geometry. Among other theories, Mochizuki introduced and developed Hodge–Arakelov theory, p-adic Teichmüller theory, the theory of frobenioids, and the etale theta-function theory.

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