1.--"My lady, take Fermat into the music room. There will be an extra spoonful of jam if you find his proof."

Tom Stoppard in Arcadia is just one of many who have helped to immortalise Fermat's Last Theorem as the Greatest Unsolved Problem of Mathematics. But last week in Jerusalem, it was Andrew Wiles, and not Arcadia's Thomassina, who was claiming that spoonful of jam. His solution of Fermat's Last Theorem was rewarded in the Knesset with one of mathematics highest accolades, the Wolf prize worth $100,000.

2.--But with its solution, have we lost the magic that this puzzle has generated over the centuries? Mathematics has benefited so much from the adoption of Fermat into the public imagination as Mathematics' Holy Grail. Fermat is probably responsible for more school children going into mathematics than any other problem. Wiles himself explained how "here was a problem that I, a 10-year-old, could understand, but none of the great mathematicians had been able to resolve. From that moment I tried to solve it myself."

3.--Could anything possibly replace Fermat's Last Theorem as Mathematics' great unsolved problem. Most people believe that mathematical research is long division to a lot of decimal places. With the advent of the computer, surely mathematics must have all been worked out by now. So is that the end of Mathematics?

4.--This perception of mathematics could not be further from the truth. Mathematics contains so many open problems, some which are much older than Fermat.

5.--Ask a mathematician to which problem they would trade their soul with Mephistopheles for a proof, then the Riemann Hypotheses has to come top of the list. It is a far greater goal for mathematics than Fermat. In fact for professional mathematicians, Fermat was a side-line. Even if Mephistopheles was to provide an uninspiring proof, the understanding that Riemann's conjecture would give us about prime numbers is immense.

6.--Although mathematicians are quite happy to explain Fermat at a dinner party, the Riemann hypothesis is a little bit more indigestible. Here though for those with a strong stomach is a flavour of what it says. The prime numbers are the indivisible building blocks of all numbers, yet their properties remain deeply mysterious. 2000 years ago Euclid showed there were infinitely many primes. This year we celebrate 100 years of knowing what proportion of all numbers are prime numbers. But if you look at a list of primes there really seems to be no nice pattern. It all looks like random noise.

7.--Around 1740, Euler identified a function (now called the Riemann zeta function) which allowed you to understand all prime numbers in one go. A function is like a computer - you feed a number in, it calculates away and gives you a number out. Those numbers which output zero are in some sense the harmonics of this function. It is these harmonics which tell you all about prime numbers. Riemann conjectured what these harmonics look like. If true, it would imply that the music of the primes is far from being just noise.

8.--So if Fermat was such a sideline compared to the likes of the Riemann hypothesis, what was all the fuss about? Fermat conjectured that, if n is a number bigger than 2, you will never be able to find three whole numbers such that the nth power of the first is the sum of the nth powers of the other two. Or, for those with a head for equations, that xn+yn=zn has no solution where x, y and z are whole numbers and n>2. (The picture shows solutions of this equation which aren't whole numbers.)

9.--Its status is not a result of its usefulness. Any mathematician would be hard pressed to find an application of this fact, even in the far reaches of pure mathematics.

10.--Part of its appeal comes from the tantalising marginal note that Fermat penned against his observation: "I have a wonderful proof of this fact which the margin is too small to contain". The famous Cambridge mathematician G. H. Hardy tried a similar trick with the Riemann Hypothesis. On a rough sea crossing, fearing for his life, he sent a joke telegram saying he had found a wonderful proof. The ship however didn't sink.

11.--Fermat was suddenly plucked from the mathematical sidelines when it became inextricably linked with a much more modish part of mathematics - elliptic curves. Ken Ribet, of the University of Berkeley, showed that Fermat would follow from a proof of a conjecture about elliptic curves named after two Japanese mathematicians Taniyama and Shimura. It was armed with this information that Wiles dedicated seven years to settling enough of Taniyama-Shimura to yield his childhood dream of proving Fermat.

12.--The Taniyama-Schimura conjecture, which Wiles partially proved, is itself just a small part of a grander vision proposed by Robert Langlands also based in Princeton. The Langlands programme as his conjecture is called, proposes to unify two seemingly different areas of mathematics - arithmetic and symmetry. It provides some sort of dictionary, translating one into another.

13.--His vision is so deep that a proof will probably not be seen in our life time and will certainly be a worthy successor to Fermat for stimulating new ideas and research. Indeed Robert Langlands was honoured jointly with the Wolf prize alongside Wiles yesterday for his extraordinary insight. In mathematics, vision is often more important than proof.

14.--But the Langlands programme, like the Riemann Hypothesis, is far too complicated to ever make an appearance in Stoppard's latest play, nor will it be drawing school children into mathematics. Perhaps as the solver of Fermat it is up to Andrew Wiles to lay down the gauntlet for the next generation.

15.--"There's no other problem that will mean the same to me." But Wiles does suggest as a candidate perhaps the oldest unsolved problem in mathematics, the so-called Problem of Congruent Numbers: find a method for determining whether a whole number occurs as the area of a right angled triangle whose sides have lengths equal to a rational number. After centuries of false proofs, Fermat himself showed that the number 1 cannot be the area of such a triangle. Some believe that Fermat thought mistakenly he could generalise his argument to prove his Last Theorem and this is what he referred to in the margin.

16.--The Problem of Congruent Numbers is simple to state and a school child can start playing round with ideas. Yet it relates to deep questions of arithmetic and has resisted centuries of attack by the best mathematicians. Perhaps a snappier name and some cryptic marginal notes by the likes of Andrew Wiles and it could find its way into the public imagination or at least a Stoppard blockbuster.

17.-- Problem of Congruent Numbers

Find a method for determining whether a whole number occurs as the area of a right angled triangle whose sides have lengths equal to a rational number. After centuries of false proofs, Fermat himself showed that the number 1 cannot be the area of such a triangle. Some believe that Fermat thought mistakenly he could generalise his argument to prove his Last Theorem and this is what he referred to in the margin.