The biggest mystery in mathematics: Shinichi Mochizuki and the impenetrable proof

[n2doc]

A Japanese mathematician claims to have solved one of the most important problems in his field. The trouble is, hardly anyone can work out whether he's right.

Davide Castelvecchi

Sometime on the morning of 30 August 2012, Shinichi Mochizuki quietly posted four papers on his website.

The papers were huge — more than 500 pages in all — packed densely with symbols, and the culmination of more than a decade of solitary work. They also had the potential to be an academic bombshell. In them, Mochizuki claimed to have solved the abc conjecture, a 27-year-old problem in number theory that no other mathematician had even come close to solving. If his proof was correct, it would be one of the most astounding achievements of mathematics this century and would completely revolutionize the study of equations with whole numbers.

Mochizuki, however, did not make a fuss about his proof. The respected mathematician, who works at Kyoto University's Research Institute for Mathematical Sciences (RIMS) in Japan, did not even announce his work to peers around the world. He simply posted the papers, and waited for the world to find out.

Probably the first person to notice the papers was Akio Tamagawa, a colleague of Mochizuki's at RIMS. He, like other researchers, knew that Mochizuki had been working on the conjecture for years and had been finalizing his work. That same day, Tamagawa e-mailed the news to one of his collaborators, number theorist Ivan Fesenko of the University of Nottingham, UK. Fesenko immediately downloaded the papers and started to read. But he soon became “bewildered”, he says. “It was impossible to understand them.”

Fesenko e-mailed some top experts in Mochizuki's field of arithmetic geometry, and word of the proof quickly spread. Within days, intense chatter began on mathematical blogs and online forums (see Nature http://doi.org/725; 2012). But for many researchers, early elation about the proof quickly turned to scepticism. Everyone — even those whose area of expertise was closest to Mochizuki's — was just as flummoxed by the papers as Fesenko had been. To complete the proof, Mochizuki had invented a new branch of his discipline, one that is astonishingly abstract even by the standards of pure maths. “Looking at it, you feel a bit like you might be reading a paper from the future, or from outer space,” number theorist Jordan Ellenberg, of the University of Wisconsin–Madison, wrote on his blog a few days after the paper appeared.

more

http://www.nature.com/news/the-biggest-mystery-in-mathematics-shinichi-mochizuki-and-the-impenetrable-proof-1.18509

[Xipe Totec]

1. So, proving that Mochizuki's proof is valid is an NP problem. Checkmate. nt

[lastlib]

2. "I find it works well enough to get me from one planet to another....."

[Jim__]

3. The abc conjecture

From wikipedia:

The abc conjecture can be expressed as follows: For every ? > 0, there are only finitely many triples of coprime positive integers a + b = c such that c > d^1+?, where d denotes the product of the distinct prime factors of abc.

To illustrate the terms used, if

a = 16 = 2⁴,
b = 17, and
c = 16 + 17 = 33 = 3·11,

then d = 2·17·3·11 = 1122, which is greater than c. Therefore, for all ? > 0, c is not greater than d^1+?. According to the conjecture, most coprime triples where a + b = c are like the ones used in this example, and for only a few exceptions is c > d^1+?.

Actually, that looks easy to prove; but wolfram puts a qualifier on ?, namely, any infinitesimal ?.