Shin Mochizuki has released his long-rumored proof of the ABC conjecture

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.<p>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 <i>anyone</i>, and he has a great nose for bullshit (as well as no patience for it).<p>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.<p>In short, outstanding people are taking this seriously.

I don't understand the new paper, but here is the ABC conjecture:<p>Among triples A + B = C, with no common factors between the three, for any given epsilon &#62; 0, there only finitely many with C &#62; 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.<p>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 &#60; product(primes(A,B,C))."<p>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:<p><a href="http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.24.5980" rel="nofollow">http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.24.5...</a><p>(Click on the PDF icon to download the paper for free.)

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

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.

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.

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:<p><pre><code> http://www.kurims.kyoto-u.ac.jp/~motizuki/top-english.html </code></pre> Clearly this man is a force to be reckoned with.