- A peer-reviewed math journal will finally publish a controversial proof of a major math idea. (But it's the mathematician's own journal.)
- Math proofs can go through many iterations and attempts before they're correct.
- The abc conjecture dates to the 1980s and is an extension of Fermat's last theorem.
Has one of the major outstanding problems in number theory finally been solved? Or is the 500-page proof missing a key piece? The verdict isn’t in yet, but the proof, at least, will finally appear in a peer-reviewed journal.
However, there’s just one catch: the mathematician himself, Shinichi Mochizuki, is one of the journal’s seniormost editors.
For those outside of academic mathematics, it’s hard to explain both how weirdly dramatic this situation has been and just how huge a successful proof of the abc conjecture would be. Nature compares it to the 1994 proof of Fermat’s last theorem, which was a gigantic landmark in math—and, at 26 years old, the most recent one on the same level of achievement.
Both proofs also involve a unique algebraic category referred to as diophantine problems. These are equations that people seek to find integer solutions for, like the special cases of the Pythagorean theorem called Pythagorean triples. When you’re studying an equation and you’re only interested in solutions that are whole numbers, this is a diophantine problem.
The abc conjecture has some commonalities with the Pythagorean theorem and other diophantine problems, involving a relationship between an a and b added together to a resulting c. Can these numbers be taken to high exponents and still have a demonstrable relationship? This is what mathematicians have been trying to prove since mathematicians first observed it in the mid 1980s. And, in fact, the conjecture is an extension of Fermat’s last theorem.
More from Nature:
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.
Mochizuki first published a Michener-novel-length proof of the abc conjecture in 2012, when he unceremoniously dumped 500 pages online and said he’d proven it. But this isn’t anyone’s first rodeo. Previous public attempts to prove the conjecture have been shown to have errors. That isn’t unusual in the process of proving complex and landmark ideas, where different scientists often iterate one new step at a time based on what their colleagues are doing.
When Mochizuki’s proof first appeared, other mathematicians reeled at both the idea of a proof of the abc conjecture and the baffling obscurity of the work itself. Mochizuki had invented a phantom scaffolding of abstract notions that shadow real mathematical ideas and notation in order to hang his very long proof upon that scaffold. In a way, trying to decipher the proof first required learning an entirely new system and notation.
To date, no one has fully understood this proof enough to validate it and communicate its structure and logical flow to others. Mochizuki himself is reclusive and hasn't really helped to illuminate his obscure mechanisms.
A few years ago, mathematicians were upset to learn that the proof was to be published in a peer-reviewed journal, and in 2018, two prominent fellow mathematicians said they were sure the proof was wrong.
The rumors of publication were false then, or perhaps rescinded after the outcry from mathematicians. But now, the proof will indeed appear in some kind of special issue of Publications of the Research Institute for Mathematical Sciences (RIMS), following what they say is the same rigorous peer review they would do for anyone. It’s not just the scandalous proof at issue here—it’s the fact that Mochizuki is RIMS’s chief editor.
Perhaps having the 500-page behemoth in print will revive the debate and bring any flaws to the surface, conclusively, once and for all. It’s hard to say for sure when the mathematics ideas themselves are so far from the norm as to seem like outsider math. Anyone who has fielded submissions for technical or mathematical publication has received gigantic, complicated documents whose authors insist they’ve proven something huge.
The truth is rarely, well, so straightforward.
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