Who will proof his proof? (Image: Research Institute for Mathematical Sciences Kyoto University)

If nobody understands a mathematical proof, does it count? Shinichi Mochizuki of Kyoto University, Japan, has tried to prove the ABC conjecture, a long-standing pure maths problem, but now says fellow mathematicians are failing to get to grips with his work.

The problem gets its name from the simple equation for adding two numbers, a + b = c, but poses deep questions about the true nature of numbers. In 2012 Mochizuki posted a 500-page paper online that claimed to solve the puzzle, but it required a dense framework of new maths dubbed “Inter-universal Teichmüller Theory” that even experienced mathematicians found difficult to follow.

Mochizuki is a highly-respected mathematician and his work is taken seriously, says Minhyong Kim of the University of Oxford, but in the years since he posted the proof no one has been able to give a definitive answer on whether it is correct. “It’s a bit disappointing that no one has come out and said it’s right or wrong,” he says.


Easy as ABC?

Now Mochizuki has posted a new report describing his efforts to explain his theory to others. He says that three researchers who studied it with his help have yet to find an error, but it will take a few more years for it to be fully confirmed.

He has also criticised the rest of the community for not studying his work in detail, and says most other mathematicians are “simply not qualified” to issue a definitive statement on the proof unless they start from the very basics of his theory.

Some mathematicians say Mochizuki must do more to explain his work, like simplifying his notes or lecturing abroad. “I sympathise with his sense of frustration but I also sympathise with other people who don’t understand why he’s not doing things in a more standard way,” says Kim. It isn’t really sustainable for Mochizuki to teach people one-on-one, he adds, and any journal would probably require independent reviewers who have not studied under Mochizuki to verify the proof.

Mochizuki’s seeming reticence calls to mind Grigori Perelman, another mathematician who refused to engage with the mathematical community and ultimately turned down a $1 million prize for his solution to a problem called the Poincaré conjecture. “This sense of stubbornness, dignity and pride is a part of what gives him the personality necessary to embark on a project like this,” says Kim. But for now, the proof remains in limbo. “It’s a curious state.”

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