# Maths genius claims prime number breakthrough The Sacks Spiral – a visual representation of the prime numbers that are the foundation of number theory.

One of the great mathematical problems of modern times appears to have been solved – but the solution is so complicated that no one can be sure it really works. How excited should we be?

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 theory, the abc conjecture is simple stuff. It is also, says mathematician Dorian Goldfeld, ‘a thing of beauty’ – ‘the most important unsolved problem in Diophantine analysis.’ 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 Grail.’

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.

## Wrong number?

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.

## You Decide

1. Can a mathematical proof really be beautiful?
2. Which tells you more about the world: maths or literature?

## Activities

1. The abc conjecture is about equations with three unknown quantities, a, b and c, such that a+b=c and such that a, b and c have no common factors. As fast as you can, find three numbers for a, b and c that satisfy those conditions.
2. The abc conjecture is about the distinct prime factors of a set of numbers. How many different prime factors are there in the numbers that make up your date of birth? Date and month should be easy. Double points for factorising the year as well.

## Some People Say...

“Numbers are not real and do not matter.”

What do you think?

## Q & A

I can’t imagine being able to prove the abc conjecture would be that useful a skill for everyday life!
It probably wouldn’t be. In fact some mathematicians are notoriously bad at managing day to day existence. But knowing advanced maths does have certain advantages.
Oh really?
Well for one thing, a strong grasp of further maths is a good way of breaking into a very well paid career in engineering or finance – or a life playing high stakes poker in Las Vegas casinos. More importantly, though, maths is what allows us to understand how the world works. In some ways, to understand the rules of maths is to understand the rules of everything.

## Word Watch

Distinct prime factors
The distinct prime factors of a number are the different prime numbers into which any larger number can be broken down. For example, the number 18 can be rewritten in prime numbers as 2 x 3 x 3. So the distinct prime factors of 18 are just 2 and 3.
Number theory
Number 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.
Diophantine analysis
Diophantine 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.
Holy Grail
In mediaeval times, mathematics and spirituality were much more closely linked. Early ‘numerologists’ believed they could use numbers to discover the true nature of God.