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Mathematical proof that rocked number theory will be published

But some experts say author Shinichi Mochizuki failed to fix fatal flaw in solution of major arithmetic problem.
Shinichi Mochizuki

Mathematician Shinichi Mochizuki.Credit: Kyoto University

After an eight-year struggle, embattled Japanese mathematician Shinichi Mochizuki has finally received some validation. His 600-page proof of the abc conjecture, one of the biggest open problems in number theory, has been accepted for publication.

Acceptance of the work in Publications of the Research Institute for Mathematical Sciences (RIMS) is the latest development in a long and acrimonious controversy over the mathematician’s proof. The journal, of which Mochizuki is chief editor, is published by Japan’s Research Institute for Mathematical Sciences (RIMS) at Kyoto University, where he works.

Two other RIMS mathematicians, Masaki Kashiwara and Akio Tamagawa, speaking in Japanese, announced the publication at a press conference in Kyoto on 3 April. The paper “will have a big impact”, said Kashiwara. Asked how Mochizuki reacted to news of the paper’s acceptance, Kashiwara said: “I think he was relieved.”

Mochizuki, who has denied requests for interviews over the years, did not appear at the press conference, and did not make himself available to reporters.

Eight years ago, Mochizuki posted four massive papers online, claiming to have solved the abc conjecture. The work baffled mathematicians, who spent years trying to understand it. Then, in 2018, two highly respected mathematicians said they were confident that they had found a flaw in Mochizuki’s proof — something many saw as a death blow to his claims.

The latest announcement seems unlikely to move many researchers over to Mochizuki’s camp. “I think it is safe to say that there has not been much change in the community opinion since 2018,” says Kiran Kedlaya, a number theorist at the University of California, San Diego, who was among the experts who had spent considerable effort trying to verify Mochizuki’s claimed proof. Another mathematician, Edward Frenkel of the University of California, Berkeley, says: “I will withhold my judgement on the publication of this work until it actually happens, as new information might emerge.”

Unsolved problem

The abc conjecture expresses a profound link between the addition and multiplication of integer numbers. Any integer can be factored into prime numbers, its ‘divisors’: for example, 60 = 5 x 3 x 2 x 2. The conjecture roughly states that if a lot of small primes divide two numbers a and b, then only a few, large ones divide their sum, c.

A proof, if confirmed, could change the face of number theory, by, for example, providing an innovative approach to proving Fermat’s last theorem, the legendary problem formulated by Pierre de Fermat in 1637 and solved only in 1994.

The saga began when Mochizuki, a respected number theorist quietly posted his preprints on 30 August 2012 — not on arXiv.org, mathematicians’ preferred repository, but on his own webpage at RIMS. Written in an impenetrable, idiosyncratic style, the papers seemed to be built entirely on mathematical concepts that were completely unfamiliar to the rest of the community — “like you might be reading a paper from the future, or from outer space”, wrote Jordan Ellenberg, a number theorist at the University of Wisconsin–Madison, on his blog soon after the papers appeared.

Mochizuki has declined all invitations to travel abroad and lecture about his work. Although, at the time, some of his close collaborators said they found the proof to be correct, experts around the world struggled, often reluctantly, to slog through it, let alone verify it. Conferences were held on the subject in subsequent years, and participants reported partial progress but said it would probably take many years to come to a conclusion. Many, including Mochizuki’s own PhD adviser, Gerd Faltings, openly criticized Mochizuki for not trying to communicate his ideas more clearly.

Then, on 16 December 2017, Asahi Shimbun, a Japanese daily newspaper, claimed that Mochizuki’s proof was close to being officially validated, an achievement that would be on par with the 1994 solution of Fermat’s last theorem.

Meanwhile, a rumour circulated that the Publications of RIMS had accepted Mochizuki’s four papers — something its editors denied at the time. But the controversy flared up again, with some mathematicians lamenting the poor perception of Mochizuki allegedly publishing in his institute’s own journal.

In December 2017, Peter Woit, a mathematical physicist at Columbia University in New York City, wrote on his blog that the journal’s acceptance would create a situation that is “historically unparalleled in mathematics: a claim by a well-respected journal that they have vetted the proof of an extremely well-known conjecture, while most experts in the field who have looked into this have been unable to understand the proof”.

Mind the gap

The rumour of imminent publication turned out to be unfounded. Then, within months, matters took a turn for the worse for Mochizuki. Two German mathematicians — Peter Scholze at the University of Bonn and Jakob Stix at Goethe University, Frankfurt — privately circulated a rebuttal of his abc proof, zeroing in on one crucial passage that they said was faulty. Scholze, in particular, is considered an authority on number theory, and would go on to win a Fields Medal — maths’ highest honour — in August 2018. In September of that year, Scholze and Stix went public, when they were quoted in an exclusive article in the maths and physics magazine Quanta, saying they had found a “serious, unfixable gap”, as Stix put it. “I think the abc conjecture is still open,” Scholze told Quanta. “Anybody has a chance of proving it.”

In comments posted on his website at the time, Mochizuki brushed aside the criticisms, hinting that the two authors had simply failed to understand his work. But several experts told Nature that much of the mathematical community considered the matter to be settled at that point.

The official acceptance of the papers now seems unlikely to change this stance. “My judgement has not changed in any way since I wrote that manuscript with Jakob Stix,” Scholze told Nature in an e-mail. (In a separate e-mail, Stix declined a request for comment.)

At the press conference, Tamagawa said the solution itself had not changed in response to Scholze and Stix’s criticism. Some comments about it will be published in the manuscript, but there will be no fundamental alteration, said Tamagawa.

If the editors of the journal “waved away these criticisms” and published the paper without major revisions, it would reflect badly on them and on Mochizuki himself, says Volker Mehrmann, the president of the European Mathematical Society (EMS), which publishes the journal on behalf of the RIMS. (The EMS has no editorial control over the journal’s content, Mehrmann says, and he was unaware that an announcement was imminent until contacted by Nature.)

But one mathematician who prefers to be quoted anonymously says that editors and referees handling these papers might have been in a nearly impossible situation. “If the best mathematicians spend time trying to work out what’s going on and fail, how can one referee on his own have any chance?”

Long and winding road to acceptance

Mathematicians often publish papers in journals for which they are editors. As long as the authors recuse themselves from the peer-review process, “such a case is not a violation of any rule, and is common”, says Hiraku Nakajima, a mathematician at the Kavli Institute for the Physics and Mathematics of the Universe in Tokyo, and formerly part of the editorial board for Publications of RIMS. Mehrmann confirms that this would not violate EMS guidelines.

Kashiwara said that Mochizuki had recused himself from the review process, and had not attended any of the editorial board meetings about the paper. The journal has previously published papers from other members of the its editorial board, he said.

Mochizuki's paper was accepted on 5 February, but a publication date has not been decided. “This is a very long manuscript and will be a special issue, so we cannot say how long it will take,” said Kashiwara.

In the world of mathematics, a journal’s seal of approval is often not the end of the peer-review process. An important result truly becomes an accepted theorem only after the community has reached a consensus that it is correct, and achieving this can take years after a paper’s official publication.

“In spite of all the difficulties over the years, I still think it would be great if Mochizuki’s ideas turned out to be correct,” says Minhyong Kim, a mathematician at the University of Oxford, UK.

Nature 580, 177 (2020)

Additional reporting by David Cyranoski.

Updates & Corrections

  • Update 03 April 2020: This story has been updated to include comments from Kashiwara and Tamagawa at a press conference in Kyoto.

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