

Shin Mochizuki has released his long-rumored proof of the ABC conjecture - bdr
http://quomodocumque.wordpress.com/2012/09/03/mochizuki-on-abc/?

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impendia
For those who don't know, Jordan Ellenberg (the author of this blog) is a
relatively famous and extremely well-regarded math professor at the University
of Wisconsin. I was fortunate enough to have him on my Ph.D. thesis committee.

Terence Tao (see the blog comments), as many HN readers know, is a Fields
medalist and quite brilliant. Brian Conrad (again, see the blog comments) is a
professor at Stanford who is incredible, he knows algebraic number theory as
well as _anyone_ , and he has a great nose for bullshit (as well as no
patience for it).

This paper is outside my immediate area, so I can't comment scientifically.
But I can speak for the culture of mathematics. Evidently, this paper is not a
bunch of obvious BS. The paper could be correct, perhaps not. It could be like
Wiles's first proof of Fermat's Last Theorem, which contained a fairly serious
gap, which was nevertheless later fixed.

In short, outstanding people are taking this seriously.

~~~
nthitz
Had him for CS 240 Intro to Discrete Mathematics, I believe. Great prof and a
great guy. He knows his stuff so this is pretty cool news!

~~~
ryanfox
He was my favorite professor and that was my favorite class in college. I
can't believe I discovered he has blog on hacker news, of all places.

If any undergrads at Madison are reading this, _take his class._ You
definitely won't regret it.

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tylerneylon
I don't understand the new paper, but here is the ABC conjecture:

Among triples A + B = C, with no common factors between the three, for any
given epsilon > 0, there only finitely many with C > R^(1+epsilon); here, R is
the "radical" of A,B,C, which is the product of the union of all primes that
divide A, B, or C.

The difference between R and the product ABC is that we take out any higher
powers of the primes. My intuitive interpretation is something like "in almost
every irreducible A+B=C, we almost have C < product(primes(A,B,C))."

Fermat's last theorem (FLT) is a major consequence. Actually, the only proof
I've seen connecting the two shows that FLT can have at most finitely many
solutions. Here's a useful survey paper:

[http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.24.5...](http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.24.5980)

(Click on the PDF icon to download the paper for free.)

~~~
prof_despistado
My intuitive interpretation is something like "in almost every irreducible
A+B=C, we almost have C < product(primes(A,B,C))."

^^Another intuitive version the conjecture is: "if A+B=C (with no common prime
factors), then it is very difficult for A, B, and C to be divisible by a prime
raised to a high power." For instance, if A was divisible by 2^1000, B was
divisible by 3^1000, and C by 5^1000, then these prime factors together would
contribute only 2 _3_ 5 = 30 to the "radical" R, which could allow C to
perhaps be much bigger than R. This can't happen "too often" (well, maybe some
finite number of times).

It was proved by Tijdeman in 1976 that the equation A + 1 = C has only
finitely many solutions where A and C are both perfect powers. Think about
this for a minute: they could be perfect squares, cubes, 4th powers... and
perhaps A = something^1000 and C = somethingelse^1001. Probably the only
obvious example of this is 2^3 + 1 = 3^2.

The ABC conjecture, if true, implies that for _any_ positive integer k, the
equation A + k = C has finitely many solutions where both A and C are perfect
powers.

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soseng
If anyone's wondering. This is Shin Mochizuki's page:
<http://www.kurims.kyoto-u.ac.jp/~motizuki/news-japanese.html>

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sadfaceunread
This paper is 4th in a sequence. There is a mistake on page 59 of the third
paper according to some other math blogs. Not sure how this may impact other
results subsequently.

~~~
ninguem2
If you are referring to comment #15 in
[http://sbseminar.wordpress.com/2012/06/12/abc-conjecture-
rum...](http://sbseminar.wordpress.com/2012/06/12/abc-conjecture-rumor-2/)
this was apparently a joke in bad taste. See the next comment.

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nicholassmith
I've just looked at this and my face melted, so could someone say why this is
important? It certainly looks impressive but it's gone over my head some what.

~~~
aaronbrethorst
Wikipedia has somewhat intelligible (to me) information on the topic:
[http://en.wikipedia.org/wiki/Abc_conjecture#Some_consequence...](http://en.wikipedia.org/wiki/Abc_conjecture#Some_consequences)

~~~
TallboyOne
I still don't particularly understand. So basically this really helps out
number theory? Where does that lead to?

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toomuchcoffee
Wow. Not only has he just demolished the ABC conjecture as a mere afterthought
to his pathbreaking research into inter-universal Teichmüller theory... he's
also an awesome Geocities-style web designer:

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

Clearly this man is a force to be reckoned with.

~~~
impendia
This is unfair. Yes, his website looks ridiculous, but it is not part of the
culture of academia to care much (and I think such websites are especially
popular in East Asia). Very few academics spend much time or effort on the
design of their website (or are impressed with the same from others). Also
keep in mind he is not a native speaker of English.

Laugh at his website if you want, but please do not infer that he's some
eccentric who has no idea what he's talking about.

~~~
toomuchcoffee
Clearly he isn't a crank, and I didn't mean for a minute to imply that he was
one. I just found the design aesthetics of his web page to be an...
interesting coincidence.

But that said -- just trust your gut instincts here. There's a long history in
mathematics of brilliant people going off into the dark for frighteningly long
periods of time, and coming back with announcements of major results that turn
out to be... interesting, but ultimately to contain irreparable gaps. And the
fact that nobody -- not even people like Tao or Conrad -- seems to know what
exactly he's talking about (even though they obviously respect the guy)
doesn't sound too encouraging, either.

~~~
tzs
Tao doesn't know exactly what Mochizuki is talking about because it is not in
any of the (many) areas that Tao is an expert in. Even Fields medalists have
their limits.

The people that do work in his area don't know exactly what he's talking about
because he's been going in a very deep and difficult direction. No one has
been following, because they've been waiting to see if he finds anything in
there. There was a good description at "Not Even Wrong":

\-----------

What Mochizuki is claiming is that he has a new set of techniques, which he
calls “inter-universal geometry”, generalizing the foundations of algebraic
geometry in terms of schemes first envisioned by Grothendieck. In essence, he
has created a new world of mathematical objects, and now claims that he
understands them well enough to work with them consistently and show that
their properties imply the abc conjecture.

What experts tell me is that, very much unlike the case of Szpiro’s proof,
here it may take a very long time to see if this is really a proof. They can’t
just rely on their familiarity with the usual scheme-theoretic world, but need
to invest some serious time and effort into becoming familiar with Mochizuki’s
new world. Only then can they hope to see how his proof is supposed to work,
and be able to check carefully that a proof is really there, not just a
mirage. It’s important to realize that this is being taken seriously because
such experts have a high opinion of Mochizuki and his past work. If someone
unknown were to write a similar paper, claiming to have solved one of the
major open questions in mathematics, with an invention of a strange-sounding
new world of mathematical objects, few if any experts would think it worth
their time to figure out exactly what was going on, figuring instead this had
to be a fantasy. Even with Mochizuki’s high reputation, few were willing in
the past to try and understand what he was doing, but the abc conjecture proof
will now provide a major motivation.

\-----------

<http://www.math.columbia.edu/~woit/wordpress/?p=5104>

