Whole numbers, addition and multiplication are among the first things schoolchildren learn, but a new mathematical proof shows that even the world鈥檚 best minds have plenty more to learn about these seemingly simple concepts.
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 鈥淎BC conjecture鈥� 鈥� and an alternative mathematical universe that should prise open many other outstanding enigmas.
To boot, Mochizuki鈥檚 proof also offers an alternative explanation for Fermat鈥檚 last theorem, one of the most famous results in the history of mathematics but not proven until 1993 (see 鈥�Fermat鈥檚 last theorem made easy鈥�, below).
Advertisement
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 Oesterl茅 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 脳 3 脳 3 脳 3 + 2 脳 2 脳 2 脳 2 脳 2 脳 2 = 5 脳 29. Simplified, the conjecture says that the large amount of smaller primes on the equation鈥檚 left-hand side is always balanced by a small amount of larger primes on the right 鈥� the addition restricts the multiplication, and vice versa.
鈥淭he ABC conjecture in some sense exposes the relationship between addition and multiplication,鈥� says of the University of Wisconsin-Madison. 鈥淭o learn something really new about them at this late date is quite startling.鈥�
Though rumours of Mochizuki鈥檚 proof started , it was only last week that he posted a series of papers on his website detailing what he calls 鈥渋nter-universal geometry鈥�, 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. 鈥淚t will be fabulously exciting if it pans out, experience suggests that that鈥檚 quite a big 鈥榠f鈥�,鈥� wrote University of Cambridge mathematician Timothy Gowers on .
Alien reasoning
鈥淚t is going to be a while before people have a clear idea of what Mochizuki has done,鈥� Ellenberg told 快猫短视频. 鈥淟ooking at it, you feel a bit like you might be reading a paper from the future, or from outer space,鈥� he .
Mochizuki鈥檚 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 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 鈥� we can talk about 鈥渢hree apples鈥� or 鈥渢hree squares鈥�, but what exactly is the mathematical object we call 鈥渢hree鈥�? 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 鈥� the empty set 鈥� 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. 鈥淧eople are aware that many natural mathematical constructions might not really fall into the universe of sets.鈥�
Terrible deformation
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 鈥渄eform鈥� basic whole numbers and push their innate relationships 鈥� such as multiplication and addition 鈥� to the limit. 鈥淗e is literally taking apart conventional objects in terrible ways and reconstructing them in new universes,鈥� says Kim.
These new insights led him to a proof of the ABC conjecture. 鈥淗ow 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,鈥� 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鈥檚 methods are completely unexpected, providing new tools for mathematical exploration.
Future megastar
Verifying the proof will itself be an endeavour, as mathematicians must comb through Mochizuki鈥檚 work line-by-line to check that the logic of each step holds true. Ellenberg expects the process of the proof will take at least a year, though any potential mistakes may be discovered sooner.
Crank claims of solving long-standing problems with esoteric methods are common in mathematics, but Mochizuki has a credible history. 鈥淗e has a terrific track record,鈥� says at the University of Montreal in Canada. Kim agrees: 鈥淭his is what makes good mathematicians take his claims very seriously, in spite of the unusual nature of the machinery he has developed.鈥�
A verified proof of the ABC conjecture would transform both Mochizuki鈥檚 career and the fundamental mathematics he spent decades studying. 鈥淢y guess is that he would become a megastar,鈥� wrote Gowers on Google+. 鈥淢ochizuki鈥檚 method, if it is found acceptable to the mathematical community, is likely to yield a completely new way of thinking about numbers,鈥� says Kim.
Fermat鈥檚 last theorem made easy
a + b = c. This basic equation sits at the very heart of the fiendish ABC conjecture 鈥� now potentially solved (see main story, above) 鈥� and links the conjecture to many other mathematical problems, including Fermat鈥檚 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 鈥� not formally posed until 1985 鈥� provides an alternative route to the theorem, and could help illuminate Fermat鈥檚 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鈥檚 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. 鈥淔ermat鈥檚 last theorem is that easy!鈥� says 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 at the University of Oxford. This could have led him to declare his own theorem, even though he hadn鈥檛 actually proved it.
Others are uncomfortable with such speculation about the ABC conjecture and Fermat. 鈥淭here is zero chance that it has anything to do with what Fermat had in mind,鈥� says of the University of Wisconsin-Madison.
