Is the ABC Conjecture finally proven? Richard Harding / Alamy Stock Photo

It is a mathematical epic five years in the making. In 2012, Shinichi Mochizuki at Kyoto University in Japan produced a proof of a long standing problem called the ABC conjecture, but no one could understand it.

Now the proof may soon be accepted for publication in a mathematical journal, which should be the end of the story – but it isn’t. Not only is the journal produced by his own university, but Mochizuki is the editor-in-chief, which could be seen as a conflict of interest.

The ABC conjecture was first proposed in the 1980s and concerns a fundamental property of numbers, based around the simple equation a + b = c. For a long time, mathematicians believe that the conjecture was true, but nobody had ever been able to prove it.


To tackle it, Mochizuki developed a whole new type of mathematics called inter-universal Teichmüller theory (IUT). His proof spanned 500 pages and baffled almost everyone who read it.

In an effort to untangle the mathematics, mathematicians have held workshops and produced a summary paper, totalling a meagre 400 pages. But while some have been converted to IUT, others have remained sceptical of the proof.

“A small number of those close to Mochizuki claim to understand the proof, but they have had little success in explaining their understanding to others,” wrote Peter Woit at Columbia University in a blog post.

Proof or not?

The fact that no journal had accepted the proof for publication had added to the cloud hanging over its validity. But according to The Asahi Shimbun, a Japanese newspaper, the proof could now be published as early as January next year.

“There has always been a rumour that the papers were submitted to a Japanese journal, which people were concerned would not give the papers enough scrutiny,” says Felipe Voloch at the University of Canterbury, New Zealand.

Though the journal is a good reputable one, the fact that it is in Japanese, from Mochizuki’s own institution, and he is the journal’s editor-in-chief means questions will still remain. “For me, the fact that it has been accepted in this journal doesn’t change much. I am still waiting for an explanation of the ideas that I can understand,” says Voloch.

Ivan Fesenko at the University of Nottingham, UK, disagrees, saying that “triple efforts” would have been applied to make sure that everything was fine before publication. And that the choice of journal can be explained by the fact that the top mathematicians in the field are Japanese. “This is an achievement on the scale we see very-very rarely in mathematics. Essentially, this is the best result in number theory in the last 50 years,” he says.

So it seems the proof remains in a precarious position. The few mathematicians who say they understand the proof will continue to champion it, whilst others will remain sceptical.

“Until there are either mathematicians who both understand the proof and are able to explain it to others, or a more accessible written version of the proof, I don’t think this proof will be accepted by the larger math community,” wrote Woit.

The status of Mochizuki’s proof has been clarified