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“ABC” proof opens new vistas in math

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You’ll have to register (free) to read the whole article, but it’s worth it. In New Scientist Jacob Aron writes:

Whole numbers, addition and multiplication are among the first things schoolchildren learn, but a new mathematical proof shows that even the world’s best minds have plenty more to learn about these seemingly simple concepts.

Shinichi Mochizuki of Kyoto University in Japan has torn up these most basic of mathematical concepts and reconstructed them as never before. The result is a fiendishly complicated proof for the decades-old “ABC conjecture” – and an alternative mathematical universe that should prise open many other outstanding enigmas.

To boot, Mochizuki’s proof also offers an alternative explanation for Fermat’s last theorem, one of the most famous results in the history of mathematics but not proven until 1993 (see “Fermat’s last theorem made easy“, below).

The ABC conjecture starts with the most basic equation in algebra, adding two whole numbers, or integers, to get another: a + b = c. First posed in 1985 by Joseph Oesterlé and David Masser, it places constraints on the interactions of the prime factors of these numbers, primes being the indivisible building blocks that can be multiplied together to produce all integers.

Dense logic

Take 81 + 64 = 145, which breaks down into the prime building blocks 3 × 3 × 3 × 3 + 2 × 2 × 2 × 2 × 2 × 2 = 5 × 29. Simplified, the conjecture says that the large amount of smaller primes on the equation’s left-hand side is always balanced by a small amount of larger primes on the right – the addition restricts the multiplication, and vice versa.

“The ABC conjecture in some sense exposes the relationship between addition and multiplication,” says Jordan Ellenberg of the University of Wisconsin-Madison. “To learn something really new about them at this late date is quite startling.”

Though rumours of Mochizuki’s proof started spreading on mathematics blogs earlier this year, it was only last week that he posted a series of papers on his website detailing what he calls “inter-universal geometry”, one of which claims to prove the ABC conjecture. Only now are mathematicians attempting to decipher its dense logic, which spreads over 500 pages.

So far the responses are cautious, but positive. “It will be fabulously exciting if it pans out, experience suggests that that’s quite a big ‘if’,” wrote University of Cambridge mathematician Timothy Gowers on Google+.

Alien reasoning

“It is going to be a while before people have a clear idea of what Mochizuki has done,” Ellenberg told New Scientist. “Looking at it, you feel a bit like you might be reading a paper from the future, or from outer space,” he added on his blog.

Mochizuki’s reasoning is alien even to other mathematicians because it probes deep philosophical questions about the foundations of mathematics, such as what we really mean by a number, says Minhyong Kim at the University of Oxford. . .

Continue reading.

The article includes this sidebar:

Fermat’s last theorem made easy

a + b = c. This basic equation sits at the very heart of the fiendish ABC conjecture – now potentially solved (see main story, above) – and links the conjecture to many other mathematical problems, including Fermat’s last theorem.

In the 17th century, Pierre de Fermat declared there were no possible solutions to the related equation, a n + b n = c n , if n is 3 or more. Maddeningly, he did not write down a proof. It was not until 1993 that Andrew Wiles found one using modern mathematics that Fermat could not possibly have known. Though many doubt Fermat even had a credible proof to back up his statement, the ABC conjecture – not formally posed until 1985 – provides an alternative route to the theorem, and could help illuminate Fermat’s line of thought.

The two puzzles are linked because if the ABC conjecture is true, it implies that there are no solutions to a n + b n = c n , if n is sufficiently large. That does not solve Fermat’s theorem outright but it vastly shortens the task. It turns the infinite problem of checking every n, in order to prove Fermat true, into a finite one. Depending on the exact formulation of the ABC conjecture, it could be that only n = 3, 4 and 5 must be checked. “Fermat’s last theorem is that easy!” says Andrew Granville of the University of Montreal, Canada.

There seems no way that Fermat could have proved ABC, but perhaps he assumed the relationships that it implies, says Minhyong Kim at the University of Oxford. This could have led him to declare his own theorem, even though he hadn’t actually proved it.

Others are uncomfortable with such speculation about the ABC conjecture and Fermat. “There is zero chance that it has anything to do with what Fermat had in mind,” says Jordan Ellenberg of the University of Wisconsin-Madison.

Written by LeisureGuy

16 September 2012 at 5:37 am

Posted in Science

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