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In a series of 4 highly technical papers, Shinichi Mochizuki claims to have proved the abc conjecture.  Though you may have never heard of it, it's a powerful conjecture in number theory that would have millions of consequences, like:

• Only finitely many numbers of the form n(n+11)(n+111) are perfect cubes.

• Only finitely many numbers of the form n! + 1111 are perfect squares.

• The equation x^a + y^b = z^c has only finitely many solutions if x,y,z are relatively prime positive integers and a,b,c are positive integers with 1/a+1/b+1/c < 1. 

Here two positive integers are relatively prime if the largest integer dividing both of them is 1.   Also, n here is a positive integer.  Also, we could replace the numbers 11, 111 and 1111 by any other integers if we wanted to!

I have no idea whether Mochizuki's proof is correct, and I couldn't understand it without a couple years of very hard work.  All I can say is that he's a serious mathematician, a Japanese algebraic geometer who has been developing Grothendieck's ideas on so-called 'anabelian geometry'.  The relevant papers are the 4 entitled 'Inter-Universal Teichmuller Theory' at the bottom of his webapge here:

http://www.kurims.kyoto-u.ac.jp/~motizuki/papers-english.html

But what does the abc conjecture actually say?    It says that for every real number p > 1,  there are only finite many triples (a,b,c) of relatively prime positive integers with a + b = c such that c > x^p, where x is the product of the prime factors of abc.

The Wikipedia article lists many consequences of this conjecture - I've just skimmed the surface.

Thanks to +Alexander Kruel and +Scott Carnahan for pointing this out.  If anyone knows the latest news on this work, let me know!
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21 comments
 
Question: Are there any known solutions to this equation?
 
"The equation x^a + y^b = z^c has only finitely many solutions if x,y,z are relatively prime positive integers and a,b,c are positive integers with 1/a+1/b+1/c < 1. "
 
Thanks for sharing, +John Baez. It's very interesting that he's working on anabelian geometry, although I guess the main thrust of his papers here is to develop a version of Teichmuller theory for elliptic curves over number fields. Note that he also claims to solve the Vojta conjecture (http://en.wikipedia.org/wiki/Vojta%27s_conjecture) for hyperbolic curves (and Szpiro's conjecture for elliptic curves, which seems to be equivalent to the abc conjecture).
 
@ Evan: Nice. I guess one thing to ask the experts is whether Mochizuki's proof might (in principle) lead to an algorithm which will list all solutions to the equation (and thus for example lead to a new proof of Fermat's last theorem, or show it's a priori trivial independent of Wiles).
 
 
+Ian Agol wrote: "guess one thing to ask the experts is whether Mochizuki's proof might (in principle) lead to an algorithm which will list all solutions to the equation..."

In number theory there's a big distinction between proofs that Diophantine equations have finitely many solutions, and 'effective' proofs that put a bound on the solutions, restricting the candidate solutions to a known finite set.  In the latter case we can in theory write a program that goes through all the candidates and finds all the solutions.  In my very brief perusal of Mochizuki's last paper, I didn't see him claiming he had an effective proof.  People usually advertise it when they have one.

But, I'm not an expert so I hope one who has carefully read the papers says what they think about this.
 
+Thomas R. Thanks for the link to that PDF file. I think it gives a good visualization of the idea of inter-universal Teichmuller theory. There are also links to his papers on the theory of "Frobenoids", which he characterizes as a "category-theoretic abstraction of the theory of divisors and line bundles on models of finite separable extensions of a given function field or number field."
 
If my understanding, backed up by a brief Googling, is correct, then the ABC conjecture implies an effective version of the Mordell conjecture. One might have thought that that means that the ABC conjecture itself has to be effective, but the links I found didn't seem to say that they needed an effective version of the ABC conjecture, which is slightly odd. But I didn't look closely, so I have probably missed something.
 
The abc conjecture is harder to write down in a popular press article than FLT.  And it doesn't come with a million-dollar prize like the Poincaré conjecture.  So I predict that it won't be as big mainstream news.
 
If I were trying to sell the ABC conjecture to a non-mathematical audience, I would do what John does above, and simply list lots of consequences, many of which are, it seems to me, of approximately equal interest to a non-mathematician as FLT, though of course without the back story. I suppose "only finitely many" isn't quite as compelling as "no", but with a bit of work I think one could get that across too -- after all, the twin-primes conjecture is easy to explain.

I suppose nobody's likely to offer a million-dollar prize for it just yet ...
 
Elliptic curves, everyone, elliptic curves... :)
 
+Thomas R. - any branch of science needs PR to get money.  Biology, chemistry, even physics do it really well.  Why not arithmetic geometry?  I don't think math should be so noble and publicity-shy that this discipline becomes as impoverished as the humanities.  My wife, working on Chinese classics, suffers such worse treatment than me... but I keep trying to convince her, and her colleagues, to explain the immense political importance of understanding Chinese classical thought.   

So, I think it would be a very good thing to popularize the consequences of the abc conjecture and explain how Mochizuki proved it using beautiful abstract ideas.   But first, people should carefully check his proof and see if it's correct!
 
+John Baez I think it's a great idea to promote a better understanding of Chinese classical thought. For one thing, there's so much material that many Chinese are, understandably, ignorant of. For another, I think it's very likely that implementing some of that classical philosophy would vastly improve the contemporary culture in China and make China a more attractive place to live in. The only snag, though, is that the general Chinese mentality is to get really tetchy when outsiders presume to inform them about things they think they know.
 
I got the impression from conversations that I've had that Japanese institutions have more say over the travel plans of their faculty, compared to, say, the situation in the US, so there may be that. Or he may just want the experts to have some time to digest his work, which is also reasonable.
 
Since Minhyong is an old pal of mine (he was a grad student at Yale when I was a postdoc there, and later we were both hanging around MIT), I should ask him what the story is. 
 
I asked him, I'll tell you what he says (unless he says not to).
 
I wonder if these results (assuming they pan out) will also advance or throw new light on the Langlands Program. Seems like they are in the same ball park.
 
What Mochizuki is claiming is a rather strong, completely effective form of ABC. (Perhaps too strong to be conceivably true? - I have a feeling it might be in contradiction with the Stewart-Tijdeman "near-misses" of ABC). If true, it will certainly yield effective Mordell, and a new proof of FLT. (And yes, effective Mordell needs an effective ABC; they are actually, in a sense, equivalent. Also, effective ABC implies an effective Roth theorem).

For a raw, explicitly effective claim from Mochizuki's fourth paper, take a look at his Theorem 1.10, which is a preliminary, restricted version of the Szpiro inequality (one form of the ABC conjecture). There, the "log(q)" on the left-hand side of the asserted inequality is precisely the logarithmic minimal discriminant log D of the elliptic curve. On the right hand side, f is the conductor N of the elliptic curve; the other term is negligible if you restrict to the number field Q. So this assertion is, essentially, an explicit form of Szpiro's inequality 1/6 log(D) < (1+epsilon) log N + const.

@John Ramsden: As far as I know, the only hint of connection between ABC and Langlands is that ABC and Taniyama-Shimura (proved by Wiles; a tiny part of Langlands) can be unified into a single conjecture: every rational elliptic curve of conductor N admits a surjective map of degree << N^{2+epsilon} from the modular curve X_0(N) of level N. In this sense, they give estimates of complementary kind: Langlands optimizes the level N, ABC optimizes the degree of the covering. 
 
I'll share that one, Nic!
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