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Titans of Mathematics Clash Over Epic Proof of ABC Conjecture (quantamagazine.org)
Two mathematicians have found what they say is a hole at the heart of a proof that has convulsed the mathematics community for nearly six years. Quanta Magazine: In a report [PDF] posted online Thursday, 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, Brian Conrad of Stanford University, 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, Conrad 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 of "Corollary 3.12" in Mochizuki's third of four papers is fundamentally flawed. The corollary is central to Mochizuki's proposed abc proof. "I think the abc conjecture is still open," Scholze said. "Anybody has a chance of proving it."
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, Conrad 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 of "Corollary 3.12" in Mochizuki's third of four papers is fundamentally flawed. The corollary is central to Mochizuki's proposed abc proof. "I think the abc conjecture is still open," Scholze said. "Anybody has a chance of proving it."
The ABC theorum is a bit hard to explain. The best I can do is taken from Wikipedia: If:
* A, B, and C are co-prime
* A + B = C
* D = the product of the unique prime factors of A, B, and C
Then D is usually not much smaller than C.
Or, put a different way, if A and B are high powers of primes, C probably isn't. For example:
* A = 64 = 2^6
* B = 81 = 3^4
* C = 145 = 5*29
* D = 870 = 2 * 3 * 5 * 29
In that example, the prime factors of C were to the first power, so D was a multiple of C. That's pretty normal.
This apparently has much broader consequences when generalized broadly to number fields, but I've never gotten my head around "primes" in fields other than integers.
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I thought theorem checking was one of the applications that AI was being touted for. Just doing a quick check, there seems to be a large number of articles (like this one, which goes back a bit: http://www.dtic.mil/dtic/tr/fu...) written about this very topic.
Rather that rely on a limited number of mathematicians, all of whom seem to know Professor Mochizuki, how about running his proof through these AI tools to see if they can validate the proof?
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