Baffling 500-page ABC maths proof to be published after eight-year row

By Donna Lu

YAY Media AS/Alamy

After a saga eight years in the making, a mathematician is finally set to formally publish a proof that rocked number theory and baffled almost everyone who read it – including other mathematicians.

In 2012, Shinichi Mochizuki at Kyoto University in Japan produced a massive proof claiming to have solved a long-standing problem called the ABC conjecture.

Spanning 500 pages across four papers, Mochizuki’s proof was written in an impenetrable style, and number theorists struggled to understand its underlying ideas.

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The work has finally been accepted in the peer-reviewed journal Publications of the Research Institute for Mathematical Sciences, but a publication date hasn’t been decided yet.

Mochizuki himself is the editor-in-chief of the journal, which is also published by Kyoto University. He hasn’t been involved in the decision to publish the proof, according to a report in Nature.

First proposed in the 1980s, the ABC conjecture is based around the equation a + b = c, and concerns the link between the addition and multiplication of integers, or whole numbers.

Simply put, it says that if a and b are made up of large powers of prime numbers – numbers only divisible by themselves and one – then c isn’t usually divisible by large powers of primes.

Mathematicians have long believed that the conjecture was true, but nobody had ever been able to prove it. Mochizuki grappled with the conjecture by developing a new type of mathematics called inter-universal Teichmüller theory.

In 2018, mathematicians Peter Scholze at the University of Bonn in Germany and Jakob Stix at Goethe University in Germany said that they had found a “serious, unfixable gap” in Mochizuki’s proof. They argued that some of Mochizuki’s reasoning was flawed and that the ABC conjecture was still an open problem.

“Opinion has definitely shifted toward the view that the proof is flawed since the letters from Scholze and Stix in 2018,” says Andrew Booker at the University of Bristol, UK. “It’s obviously bad for the [number theory] community if the result is declared a theorem in some circles but not others.”

At a press conference on Friday in Kyoto announcing the paper’s acceptance, which Mochizuki did not appear at, mathematician Akio Tamagawa said the proof included no fundamental changes in response to Stix and Scholze’s criticism.

“The closest we’ve come to this sort of dilemma in recent times is the controversy surrounding Thomas Hales’s proof of the Kepler conjecture in 1998,” adds Booker. “It was the other way around in that case, in that the proof was viewed as impenetrable but probably correct.” Hales’ proof was eventually formally verified with the aid of a computer in 2014.

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