When the British mathematician Andrew Wiles told the world about his proof of the Last Theorem of the seventeenth century French lawyer, Pierre de Fermat, it looked as if the Holy Grail had been grasped.

Fermat's Last Theorem has often been called the greatest unsolved riddle of mathematics. But many mathematicians would argue that this name belongs rather to an idea first put forward in the middle of the nineteenth century by the German mathematician Bernhard Riemann: The Riemann Hypothesis.

It remains unresolved but, if true, the Riemann Hypothesis will go to the heart of what makes so much of mathematics tick: the prime numbers. These indivisible numbers are the atoms of arithmetic. Every number can be built by multiplying prime numbers together. The primes have fascinated generations of mathematicians and non-mathematicians alike, yet their properties remain deeply mysterious. Whoever proves or disproves the Riemann Hypothesis will discover the key to many of their secrets and this is why it ranks above Fermat as the theorem for whose proof mathematicians would trade their soul with Mephistopheles.

Although the Riemann Hypothesis has never quite caught on in the public imagination as Mathematics' Holy Grail, prime numbers themselves do periodically make headline news. The media love to report on the latest record for the biggest prime number so far discovered. In November 1996 the Great Internet Prime Search announced their discovery of the current record, a prime number with 378,632 digits. But for mathematicians, such news is of only passing interest. Over two thousand years ago Euclid proved that there will be infinitely many such news stories, for the primes never run dry.

Rather mathematicians like to look for patterns, and the primes probably offer the ultimate challenge. When you look at a list of them stretching off to infinity, they look chaotic, like weeds growing through an expanse of grass representing all numbers. For centuries mathematicians have striven to find rhyme and reason amongst this jumble. Is their any music that we can hear in this random noise? Is there a fast way to spot that a particular number is prime? Once you have one prime, how much further must you count before you find the next one on the list? These are the sort of questions that have tantalized generations.

Last year we celebrated the centenary of perhaps the deepest fact known to date about the prime numbers since Euclid. Proved by the French mathematician Hadamard and the Belgian mathematician de la Vallee Poussin, the Prime Number Theorem tells us how many primes we should expect to find between 1 and any number N. For example in the first 100 numbers, it says that there should be 25 primes. But in the first million, only one in fifteen should be prime. It therefore gives the probability that a number will be a prime and it further says that as the says that as numbers get bigger this probability gets smaller. So the primes thin out, getting rarer and rarer. Euclid guarantees that there will always be more primes to find. On the other hand, the Prime Number Theorem tells you how rare they become.

The theorem gives us a formula for how many primes we should expect to find less than any number N. But it is not an exact formula. And this inexactness is at the heart of the Riemann Hypothesis. There is an analogy here with coin tossing. If I toss a coin a million times, then with a fair coin we should get half heads and half tails. But we don't expect to get exactly five hundred thousand heads. By the nature of this being a random process we will not be surprised by a variation of about one thousand either side of this number.

The Riemann Hypothesis would say that looking for primes is rather like tossing a coin. We have a one hundred year old formula which tells us roughly how many primes we should expect to find. But we know that this is not an exact formula. Riemann predicted that the error term in this formula is the same as the error we expect to see when tossing coins making the primes look in some sense like a random process. So given a prime, to find the next prime on the list will be like waiting for the next head to appear when tossing a coin. Given that we can't expect an exact formula, this distribution of the primes conjectured by Riemann is as nice a one as we could hope for.

The Riemann Hypothesis says that there aren't any mysterious patterns that we haven't already discovered among the primes. If it were false it would imply that there was some structure in the primes that we had missed over the centuries.

To prove his prediction, Riemann showed that you need to move the goal post and prove a result in a seemingly unrelated area called complex function theory. These strange connections are one of the great themes of mathematics. Riemann identified a function which encoded the structure of the primes. A function is like a computer - you feed it a number and it outputs another number. In Riemann's function, called the Riemann zeta function, you actually feed in two numbers which define the co-ordinates of points on a flat piece of paper. We can therefore get a graphical representation of this function as a surface sitting above the piece of paper where the height of the surface above a point on the paper is the output of the function at that point. Figure 1 and 2 show two different pictures of this surface. (The picture should actually live in four dimensions but the output of the function has been doctored for our humble 3-D consumption.)

It was Riemann's great insight that the behaviour of prime numbers, by their nature discrete objects where you have to jump from one to the next, should be connected with a smooth continuous surface like the zeta function. Nevertheless they are inextricably linked.

One of the important features of a function are those numbers where the function outputs zero. These are like the harmonics of your function. For example, if one remembers the picture of the sine or cosine functions that one learns in school, the picture oscillates outputting zero at regular intervals.

For Riemann's zeta function, these harmonics have an extra significance for they describe the sound of the primes.Therefore anything you can say about these harmonics should tell you significant things about the prime numbers. As we have drawn our picture the peaks actually represent the zeros or harmonics. You might notice that these first few peaks all seem to lie in a straight line. Riemann predicted that in fact all the points at which this function is zero should lie in a straight line where one of the co-ordinates is always 1/2. What is rather startling is that this very regimented behaviour of the harmonics implies the coin-tossing nature of the primes. This is in fact the conjecture that Riemann originally proposed and what mathematicians refer to as the Riemann Hypothesis: the zeros of the Riemann zeta function lie on a straight line.

What evidence is there then for this conjecture being true? Riemann's papers show that he had quite sophisticated methods to detect zeros which led him to his conjecture. Unfortunately many of his writing were destroyed so it is unknown how much Riemann actually knew. Perhaps we might find his own marginal comment of a wonderful proof to rival Fermat's historic tease. The famous Cambridge mathematician G.H. Hardy in the twenties proved that infinitely many of these zeros lie in a line. He almost went on to provide the Riemann Hypothesis with a story to equal Fermat's cryptic note in the margin. On a rough sea crossing fearing for his life, he sent a joke telegram saying that he had found a wonderful proof of the Hypothesis. The ship, however did not sink.

Since the arrival of the supercomputer it has been possible to test the conjecture way beyond Riemann's wildest dreams. It is now known that the first one and a half billion zeros lie on this straight line. However such convincing evidence can be misleading. At AT&T Bell Labs Andrew Odlyzko together with Hermann te Riele proved that a related conjecture was in fact false despite similar overwhelming evidence. Since there are known to be infinitely many zeros, the computer will never be able to tell us conclusively that they all lie in a straight line. Instead we must rely on human insight for a solution, a relief perhaps to those unnerved by the victory of IBM's Deeper Blue over the chess Grand Master Gary Kasparov.

It may seem surprising to find a commercial organisation like AT&T interested in the Riemann Hypothesis. However prime numbers are no longer just the plaything of the mathematician but hold the key to the world's finances. The codes that protect our credit card numbers when we send them across the internet depend on mathematical puzzles about prime numbers which at present are too difficult for us or a computer to crack. However the insight that a proof of the Riemann Hypothesis would give us about the primes could be enough to make the secrets of today, tomorrow's public property. That is why you find the likes of AT&T research laboratories dedicating time and money to monitoring progress in the ivory towers of academia.