Not as easy as ABC Bernhard Classen/Alamy Stock Photo

A summary of a massive mathematical proof that has baffled mathematicians for the past five years may help a few more people get to get grips with the key ideas. How long is the explainer? A mere 300 pages.

The original proof is of the long-standing ABC conjecture that explores the deep nature of numbers, centred on the simple equation a + b = c. It has been thought for some time that the conjecture is true, and in 2012, Shinichi Mochizuki at Kyoto University in Japan produced a proof to settle the matter.

Unfortunately, it was 500 pages long and developed a whole new type of mathematics called inter-universal Teichmüller theory (IUT) that nobody at the time could really understand.


Since then, two conferences have tried to get to grips with the work, and some mathematicians have made progress. “In 2012 there were no experts on IUT. By now, partially as the result of our conferences, the number of experts is between 10 and 20,” says Ivan Fesenko at the University of Nottingham, UK, who helped organise the events.

Because IUT is so different from other mathematical approaches, much of the language is unfamiliar to mathematicians. Mochizuki refuses to travel outside Japan to help explain his work, and his written explanations can seem impenetrable.

More clarity?

To help clear things up, Go Yamashita, a colleague of Mochizuki at Kyoto University, has written a 300-page summary paper that tries to clarify some of the language.

“I’ve only taken a superficial glance at the document, but my overall impression is positive,” says Minhyong Kim at the University of Oxford. “The language strikes me as substantially more accessible than that of the original papers.”

But others are not so sure. “I don’t expect it will help clear up the matter. It’s very much in the same style of Mochizuki’s writing. Mochizuki and his group can’t seem to communicate and nobody from outside has had any success in understanding the details,” says Felipe Voloch at the University of Canterbury in Christchurch, New Zealand.

For the time being, the ABC conjecture is likely to remain in limbo, as no journal has been willing to publish it – the final stamp of approval for a proof. “Usually in pure mathematics, it’s sufficient that just one referee understands and approves a paper for it to be published,” says Fesenko. “In this case, we have more than 10 people. Anyone else can study it if they want.”