Wrong number?

Mathematicians around the world are in a state of high excitement this week, after a researcher at the University of Kyoto claimed to have found a solution for one of the hardest and most important problems in modern mathematics: the abc conjecture. If the solution works, this would be the greatest mathematical breakthrough of the 21st Century so far.

For non-mathematicians, even understanding exactly what the abc conjecture is can be quite a challenge. Perhaps the fastest simple way to put it is this: for any equation, a + b = c, where a, b and c are positive integers (whole numbers) with no common factors, if you multiply together the distinct prime factors of a, b and c, and then raise that number to any power greater than one, the result will be greater than some minimum value higher than zero.

To most people, the conjecture might sound both very complicated and slightly boring. But for those who understand the mysteries of number theoryNumber theory is one of the oldest branches of mathematics. It deals with the behaviour and characteristics of whole numbers, above all, the prime numbers. At the moment, there is still no fundamental theory that can predict the appearance of primes., the abc conjecture is simple stuff. It is also, says mathematician Dorian Goldfeld, 'a thing of beauty' - 'the most important unsolved problem in Diophantine analysisDiophantine equations are a class of equations named after an Ancient Greek mathematician, Diophantus of Alexandria, who lived in the Third Century AD. Diophantine problems have been studied for hundreds of years, but many answers remain extremely difficult to find..' In this ancient subfield of number theory, the abc conjecture works like a sort of skeleton key. If it proves to be true, a whole series of other impenetrable Diophantine problems can, at last, be cracked open.

That, however, remains a big 'if'. Shinichi Mochizuki, one of Japan's top mathematicians, claims to have proved that the abc conjecture is true - but Mochizuki's 'proof' is fully 500 pages long. At the moment, no one in the world can fully understand it, and it could take months of hard labour for anyone to learn enough to say whether what he says is true.

Nonetheless, there are plenty of other mathematicians willing to try. A working proof of the abc conjecture is about as close as modern mathematics gets to having a 'Holy GrailSomething that is eagerly sought after. The real holy grail was the cup that Jesus used at the Last Supper, for which Christians searched fruitlessly for centuries..'

But mathematical true believers will say this is beside the point. What makes the abc conjecture exciting is not that it is useful, but that it is beautiful. If it has really been proved true, mathematicians will have gained a dramatic new insight into the fundamental language of the universe.

While mathematicians celebrate, many others will be scratching their heads in confusion. These numbers and equations seem impossibly far away from anything that could ever matter in the real world.

That is not quite true, as IT experts could point out. Number theory is what builds the codes that keep bank details and passwords safe for shoppers online. The abc conjecture could lead to real advances in secure communications.