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Math Titans Clash Over Epic Proof of the ABC Conjecture

abc conjecture, illustration

Credit: Quanta Magazine

In a report posted online, Peter Scholze of the University of Bonn and Jakob Stix of Goethe University Frankfurt describe what Stix calls a "serious, unfixable gap" within a mammoth series of papers by Shinichi Mochizuki, a mathematician at Kyoto University who is renowned for his brilliance. Posted online in 2012, Mochizuki's papers supposedly prove the abc conjecture, one of the most far-reaching problems in number theory.

Despite multiple conferences dedicated to explicating Mochizuki's proof, number theorists have struggled to come to grips with its underlying ideas. His series of papers, which total more than 500 pages, are written in an impenetrable style, and refer back to a further 500 pages or so of previous work by Mochizuki, creating what one mathematician has called "a sense of infinite regress."

Between 12 and 18 mathematicians who have studied the proof in depth believe it is correct, wrote Ivan Fesenko of the University of Nottingham in an email. But only mathematicians in "Mochizuki's orbit" have vouched for the proof's correctness, Brian Conrad of Stanford University commented in a blog discussion last December. "There is nobody else out there who has been willing to say even off the record that they are confident the proof is complete."

Nevertheless, wrote Frank Calegari of the University of Chicago in a December blog post, "mathematicians are very loath to claim that there is a problem with Mochizuki's argument because they can't point to any definitive error."

That has now changed. In their report, Scholze and Stix argue that a line of reasoning near the end of the proof in Mochizuki's third of four papers is fundamentally flawed. The corollary is central to Mochizuki's proposed abc proof.

From Quanta Magazine
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