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

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. 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.

Three years on, Mochizuki’s proof remains in mathematical limbo — neither debunked nor accepted by the wider community. Mochizuki has estimated that it would take an expert in arithmetic geometry some 500 hours to understand his work, and a maths graduate student about ten years. So far, only four mathematicians say that they have been able to read the entire proof. Source: The biggest mystery in mathematics: Shinichi Mochizuki and the impenetrable proof

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

  1. Phil Krause says:

    Easy as ABC, he should have solved something useful like the salesman problem. He could have made himself rich at the same time.

  2. Phil Krause says:

    The traveling salesman problem, is one been debated by many people for many years. There is a million $ prize on offer for the first to solve it. Basically it’s to find the shortest distance it takes a salesman to travel around a number of towns returning to his home. The only way to solve 100% is to calculate every possible route. This is ok for a few places but it increases very fast with the number of towns visited. This problem also applies to many other fields besides salesmen. It doesn’t sound very complicated but it must be because so much effort has gone into it for many years; hence the prize.

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