Mathematician claims to have proved the 'abc conjecture'

Shinichi Michizuki claims to have proved the famous hypothesis; his peers say he may have arrived at a 'grand unified theory of numbers'

PUBLISHED : Sunday, 23 September, 2012, 12:00am
UPDATED : Monday, 24 September, 2012, 10:21am

This summer, the international mathematical community was abuzz with rumours that a proof of the famous "abc conjecture", a key problem in number theory, had been found.

Then, early this month, and without fanfare, Shinichi Mochizuki, a widely respected mathematician at Kyoto University, posted four papers totalling 500 pages on his website claiming to have proved it. If his proof is valid, it would be an achievement on a par with the proofs of Fermat's last theorem by the Briton Andrew Wiles in 1995 and the Russian Grigory Perelman's Poincaré conjecture, in 2003. So, what's all the fuss? For laypeople, the first step of the conjecture may seem absurdly simple: a + b = c, or as an example, 1 + 2 = 3.

But for mathematicians, a valid proof will hold deep insights into the nature of integers and prime numbers, and so will have repercussions in practically every important corner of number theory. Some have dubbed it "the grand unified theory of numbers". Fermat's last theorem - according to which an + bn =cn has no interger solutions if n > 2 - is just one of many important results that will follow as a direct consequence of the abc conjecture.

The problem is, as The New York Times reported, even top mathematicians have no idea whether Mochizuki's proof is valid, or even what he is doing. This is because he has invented whole new concepts and notations. For now, he has kept quiet, preferring to let his peers work through his papers towards a definitive consensual judgment.

In his influential blog quomodocumque, Jordan Ellenberg, a maths professor at the University of Wisconsin, Madison, said he hoped the proof was right but predicted it would take many months for professional mathematicians to work it out.

"I hope it's true: my sense is that there's a lot of very beautiful, very hard math going on in Shin's work which almost no one in the community has really engaged with, and the resolution of a major conjecture would obviously create such engagement very quickly," he wrote. "Well, now the time has come. I have not even begun to understand Shin's approach to the conjecture. But it's clear that it involves ideas which are completely outside the mainstream of the subject. Looking at it, you feel a bit like you might be reading a paper from the future, or from outer space."

The abc conjecture, first proposed independently by the Briton David Masser and Frenchman Joseph Oesterle, in 1985, considers a simple equation of three integers, a + b = c, where the three do not share any common divisors other than 1. Examples are: 1 + 2 = 3 and 81 + 64 = 145.

Here's a more formal way of stating it, according to a paper by Ivars Peterson on the website of the Mathematical Association of America. You need to remember some basic definitions. A square number is the product of some integer with itself, or n x n. Prime numbers are those that can only be evenly divided by 1 and themselves, such as 5 and 17; and the prime factors of a positive integer are the prime numbers that divide that integer without leaving a remainder. For example, the prime factors of 6 are 2 and 3, as both numbers are primes.

The abc conjecture involves the concept of square-free numbers, or numbers that cannot be divided by a square number. The square-free part of a number, n, denoted by sqp(n), is the largest square-free number that can be obtained by multiplying the prime factors of n. The abc conjecture makes a statement about pairs of numbers that have no prime factors in common, Peterson explained. If a and b are two such numbers and c is their sum, the abc conjecture holds that the square-free part of the product abc, denoted by sqp(abc) and divided by c, is always greater than 0. And, sqp(abc) raised to any power greater than 1 and divided by c is always greater than 1.

Got it? No? Don't worry, this is just to give you a feel for what maths geeks worry about when they talk about the conjecture.

As quoted by Peterson, Andrew Granville, currently of the University of Montreal, said: "The abc conjecture is amazingly simple compared to the deep questions in number theory, but this strange conjecture turns out to be equivalent to all the main problems. It's at the centre of everything that's been going on. If you're working on a problem in number theory, you often think about whether the problem follows from the abc conjecture."

Columbia University mathematician Dorian Goldfeld was quoted as saying: "It is more than utilitarian. Seeing so many problems unexpectedly encapsulated into a single equation drives home the feeling that all the sub-disciplines of mathematics are aspects of a single underlying unity, and that at its heart lie pure language and simple expressibility."

You can begin to see why mathematicians are all excited now about Mochizuki. But it will take some time before they know whether he has nailed it or just sounded a false alarm.