As children we all learn our abc’s; as adults very few ponder the ABC Conjecture in mathematics. The first is often a simple task of rote memorization; the second is a troublesome mathematical problem with a fiendishly complex solution (maybe).
?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.
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+.
“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. The early 20th century saw a crisis emerge as mathematicians realised they actually had no formal way to define a number – we can talk about “three apples” or “three squares”, but what exactly is the mathematical object we call “three”? No one could say.
Eventually numbers were redefined in terms of sets, rigorously specified collections of objects, and mathematicians now know that the true essence of the number zero is a set which contains no objects – the empty set – while the number one is a set which contains one empty set. From there, it is possible to derive the rest of the integers.
But this was not the end of the story, says Kim. “People are aware that many natural mathematical constructions might not really fall into the universe of sets.”
Rather than using sets, Mochizuki has figured out how to translate fundamental mathematical ideas into objects that only exist in new, conceptual universes. This allowed him to “deform” basic whole numbers and push their innate relationships – such as multiplication and addition – to the limit. “He is literally taking apart conventional objects in terrible ways and reconstructing them in new universes,” says Kim.
These new insights led him to a proof of the ABC conjecture. “How he manages to come back to the usual universe in a way that yields concrete consequences for number theory, I really have no idea as yet,” says Kim.
Because of its fundamental nature, a verified proof of ABC would set off a chain reaction, in one swoop proving many other open problems and deepening our understanding of the relationships between integers, fractions, decimals, primes and more.
Ellenberg compares proving the conjecture to the discovery of the Higgs boson, which particle physicists hope will reveal a path to new physics. But while the Higgs emerged from the particle detritus of a machine specifically designed to find it, Mochizuki’s methods are completely unexpected, providing new tools for mathematical exploration.