The sums of his ambition

Teenager Arran Fernandez's brilliance has made headlines since he was five. Now he is likely to be the youngest undergraduate at Cambridge University with his sights set on solving one of the greatest mathematical conundrums.

Arran Fernandez, aged 7 from Surrey, plays on a Comptometer (a 1920's accounting device), after receiving his intermediate tier GCSE certificate in maths to make him the youngest person to pass the exShow more

When Arran Fernandez was five years old, he said he wanted to be one of three things when he grew up. Two were pretty predictable: astronaut and lorry driver. Now, at the age of 14, he has made headlines by choosing his very unconventional third option: mathematician. Arran has just been offered a place to read maths at the University of Cambridge and, if he gets the right grades, this summer, he'll become its youngest undergraduate since William Pitt, the Younger turned up there over 230 years ago.

There is no doubting Arran's exceptional abilities. Taught by his parents at home in Surrey, England, he passed his first formal examinations in maths at the age of just five (the usual age is 16), an achievement that prompted his first exposure to the media. He has since passed all his advanced maths exams, years ahead of time. Even so, stories of such prodigies entering university while barely out of short trousers provoke mixed responses from professional mathematicians. Quite apart from the wisdom of exposing someone so young to student life, there is the question of coping with the notoriously big jump in standard between school maths and the subject as taught at university.

In any case, there are any number of bright children with a facility for passing exams. To merit the epithet of mathematical prodigy, one needs something more: creativity if not outright genius. In Arran's case, the signs are that he is no mere hot-housed nerd. At the age of five, he had already published some new, albeit minor, discoveries concerning number sequences. But something he said at the press conference after his acceptance by Cambridge may prove even more significant. He declared that he would very much like to prove the Riemann Hypothesis.

That might mean little to most people, but to professional mathematicians it shows that Arran knows about serious problems far beyond school exams. For the Riemann hypothesis is one of the greatest unsolved conundrums in mathematics, and a worthy target for the attention of any would-be prodigy. As it happens, the origins of the Riemann Hypothesis lie in an apparently trivial discovery made by another teenage mathematician over 200 years ago. It centres on the so-called prime numbers, those which can only be divided exactly by either themselves or one. The primes are the building blocks of every ordinary number, which is either prime itself or a unique product of two or more primes.

In Germany in 1792, a 15-year-old mathematical prodigy named Carl Friedrich Gauss discovered a curious "law" that seemed to describe the distribution of primes among ordinary numbers. At first sight, they seem to pop up at random, albeit progressively infrequently. Of the numbers between one and 100, 25 per cent are primes; between 100 and 1000, however, their prevalence has fallen to around 17 per cent.

Gauss's formula described this pretty well - but not perfectly. For some reason, it always seemed to overestimate the true number of primes. Unable to explain why, Gauss moved on to other matters, and ended up being regarded as one of the greatest mathematicians of all time. Even so, he never forgot about his youthful discovery, and returned to it years later. Pushing the law out to ever larger primes, Gauss decided that his law would always overestimate the number of primes. Yet for once, his intuition let him down: at the start of the 20th century mathematicians proved that his formula eventually starts to underestimate the true number of primes. Exactly where, however, remains unknown; computer searches have so far failed to find the switch-over point. What is clear is that there remain huge holes in our understanding of prime numbers.

This is where Riemann's hypothesis - and perhaps, one day, Arran - comes in. In 1859 the German mathematician Bernhard Riemann found an unexpected connection between primes and so-called "complex numbers", a kind of extended version of ordinary numbers. Riemann quickly exploited the breakthrough to derive a new and incredibly accurate version of Gauss's old formula. Since then, other mathematicians have used Riemann's work to extract a host of other insights about the properties of numbers. But there is a problem: everything rests on a complex formula found by Riemann behaving in a very specific way.

Virtually every mathematician believes it does so, but they don't know for certain. Proof of this so-called Riemann hypothesis has so far eluded everyone. If he succeeds, he will win a permanent place in the pantheon of mathematicians - plus one of the $1 million prizes offered by an American millionaire, Landon Clay, for solving the outstanding problems of mathematics. But we may all have reason to celebrate. Prime numbers lie at the heart of encryption software used to protect communications and money transfers over the internet. Yet the security of these methods has never been proved. Mathematicians suspect that efforts to demonstrate the truth of the Riemann hypothesis will lead to new techniques able to resolve these nagging doubts.

So what are Arran's chances of success? Not huge, but there are certainly precedents. In 1963, a 10-year-old English boy named Andrew Wiles decided he was going to tackle another long-standing mathematical mystery, known as Fermat's Last Theorem. This states that it is impossible to find two whole numbers which when raised to any power greater than two and added together give another whole number raised to the same power.

After studying maths at both Oxford and Cambridge universities, Wiles was told to forget about his childhood dream and focus on another, apparently unrelated branch of mathematics. Years later Wiles realised there was a completely unexpected connection between his work and Fermat's Last Theorem, which led him to a now celebrated proof, published in 1995. Perhaps Arran will one day be at the centre of another press conference, this time brandishing the proof of the mathematical puzzle that defeated his illustrious 18th century forebear.

Robert Matthews is Visiting Reader in Science at Aston University, Birmingham, England