ABC Proof: Japanese Mathematician Solved a Problem So Complicated No One Can Check His Work

A mathematician will soon publish his answer to an infamous conjecture, but it's so complicated no one can check if he's right. Shutterstock

Updated | Five years ago, Japanese mathematician Shinichi Mochizuki claimed to have created a proof for a notoriously complex problem called the ABC conjecture, but none of his peers could figure out if it was right. Now, according to Japanese news site The Asahi Shimbun, he's preparing to finally publish his work in the journal of the Kyoto University's Research Institute for Mathematical Sciences—of which he's the editor. The publication date could be as soon as January 2018.

In the 1980s, two European mathematicians posed a baffling conjecture. The problem begins with one of algebra's most basic premises, adding two positive whole numbers to equal a third one: a + b = c. From there, though, everything goes a bit sideways. The ABC conjecture questions the nature of numbers themselves. And although, according to New Scientist, many mathematicians believed the equation was true, no one was able to prove it.

5 yrs later, experts still can't understand Mochizuki's 500-page alleged proof of the ABC conjecture. Now it's to be published — but in a journal edited by Mochizuki. The situation is "historically unparalleled in mathematics," says @notevenwrong.

— Natalie Wolchover (@nattyover) December 18, 2017

In 2012, in a series of four papers totaling about 500 pages posted to his website, Mochizuki proposed his new inter-universal Teichmüller theory, which according to New Scientist is basically an entirely new kind of math. Other mathematicians have spent the years since trying to decipher it. Scores of conferences and papers hundreds of pages long have been devoted to figuring out what the unprecedented theory actually meant, and still no one was really able to get to the bottom of it.

"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," Columbia University mathematician Peter Woit wrote in a Columbia blog post. "The usual mechanisms by which understanding of new ideas in mathematics gets transmitted to others seem to have failed completely in this case."

Meanwhile, Mochizuki's frustration at the holdup continued to grow.

Mochizuki is universally described as a reclusive prodigy, and according to The Asahi Shimbun, he spent a decade working in isolation to develop his theory. Though he's highly respected, and currently highly in demand, Scientific American reported that because he dislikes travel he's unwilling to discuss his theory at any lectures outside Japan. This means that there aren't that many mathematicians who have even read his papers, let alone tried to decipher them.

I have deleted my earlier tweet which I wrote being unaware that S.Mochizuki is Editor-in-Chief of the journal to which he submitted his papers. This is unfortunate. It creates the appearance of a conflict of interest & hence undermines one's confidence in the refereeing process.

— Edward Frenkel (@edfrenkel) December 16, 2017

The peer-review process is the check-and-balance system that keeps developments in math and science accountable. Advancing a theory without it is irresponsible in the eyes of academics. The Kyoto University RIMS journal is held in high regard, and the fact that Mochizuki is the editor doesn't necessarily mean the paper won't be subject to the same kind of rigorous review it would receive anywhere else. But his route to publishing is definitely unusual, and many mathematicians find it telling that the proof wasn't accepted anywhere else.

"There has always been a rumor that the papers were submitted to a Japanese journal, which people were concerned would not give the papers enough scrutiny," Felipe Voloch, a mathematician at the University of Canterbury, New Zealand, told New Scientist. "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."

This article has been updated to include a quote from Peter Woit.