Here's a challenge for you: name a mathematician.

If that was too easy, or if you had to cast your mind back many years to come up with that scourge of your schooldays, Pythagoras, let me make things more difficult: name a living mathematician.

At this point you may likely think of Stephen Hawking, who is actually a theoretical physicist; or you may be drawing a complete blank.

If the latter, that's not surprising. Mathematicians vary widely in temperament and lifestyle, so that, for example, John von Neumann, possibly the twentieth century's most versatile mathematician, was a real social animal whose home was famous for its parties; while his Princeton contemporary, Kurt Godel, the greatest mathematical logician of the century, was a social recluse who ended his life when he stopped eating and starved to death.

Either way, mathematics is not a subject that normally leads to fame and glory, and the world's most famous mathematician will usually be less well-known than the backup quarterback for any major football team. So when, in 1993, the name of the mathematician Andrew Wiles made the front page of newspapers around the world, it was clear that he had accomplished something spectacular.

What Wiles, born in England and now living in America, had done is solve a problem known as Fermat's Last Theorem. That theorem is unusual in a number of ways. First, it makes a statement about the possible whole-number solutions of a certain equation that is simple enough to be understood by people with a minimum of mathematical training.

Second, the theorem had been around for a very long time. It was discovered and stated by Pierre de Fermat in about 1640. For the next three and a half centuries, the world's best mathematicians tried in vain to prove or disprove Fermat's statement. To make matters particularly galling, Fermat wrote out his theorem in the margin of a textbook by another author. Here it is (translated; Fermat, like all scientists of his time, wrote in Latin): "It is impossible for a cube to be written as a sum of two cubes or a fourth power to be written as the sum of two fourth powers or, in general, for any number which is a power greater than the second to be written as a sum of two like powers."

If Fermat had stopped there, it wouldn't have been too bad. But he added a remark that has ever since driven mathematicians to distraction: "I have a truly marvelous demonstration of this proposition, which this margin is too narrow to contain."

Did Fermat have a proof, or didn't he? No one knows. What is beyond question is that he was one of the great mathematicians of all time. What is also beyond question is that if Fermat did have a proof, it could have been nothing like the one provided by Andrew Wiles in 1993. Wiles called upon a range of mathematical tools that simply did not exist before the twentieth century. This is not intended to minimize in any way his achievement. He worked on the problem, with great intensity and in essential isolation, for seven years, and had to create new mathematical tools along the way. Wiles fully deserved to be, for a brief period, as famous as a winning Super Bowl quarterback. It is also absolutely certain that his name will be well known to mathematicians hundreds of years from now.

This leaves an interesting question. If for hundreds of years the proof of Fermat's Last Theorem was a kind of Holy Grail of mathematics, what comes next? Are there other longstanding problems that provide a continuing challenge?

Fortunately, there are many. A good number of them involve primes, numbers that can't be written as a product of other numbers (so, for example, 13 is a prime number, while 14 = 2x7 is not). Here is a deceptively simple problem, known as the Goldbach Conjecture: Every even number can be written as the sum of two prime numbers. The Goldbach Conjecture has been around since 1742. It has been tested and found to be true for every number up to 100,000,000; but examples, no matter how numerous, can never add up to an actual proof. Some progress has been made. For example, in 1937 the Russian Vinogradov proved that every sufficiently large number can be written as the sum of three prime numbers. But the final proof of the Goldbach Conjecture is still lacking.

The most important unsolved problem in mathematics, however, is almost certainly the Riemann Conjecture. You might ask, what do I mean by "important"? Pure mathematics has no practical uses, and this uselessness has even been a boast of mathematicians.

The importance of a theorem in mathematics is measured by the degree to which other mathematical work depends upon it. The Riemann Conjecture is uniquely central in this way. Numerous papers today begin "Assuming that the Riemann Conjecture is correct, then we have the following results..."

So what is the Riemann Conjecture? Fermat's Last Theorem was easy to state in simple terms. The Riemann Conjecture isn't. It proposes that almost all the zeroes of a particular mathematical function, known as the Riemann Zeta Function, lie along a certain line of the complex plane; and I am sorry, but I know of no less technical way to state it.

The Riemann Conjecture is a relative newcomer compared with Fermat's Last Theorem or the Goldbach Conjecture. It was proposed by Bernhard Riemann in 1859; again, progress has been made in the past century. For instance, in 1914 the English mathematician G.H. Hardy showed that an infinite number of zeroes of the Riemann Zeta Function lie on the specified line. Again, that falls short of the necessary proof.

The Riemann Conjecture stands as a constant challenge to every mathematician in the world. Whoever proves (or disproves) it will earn a sure place in mathematical history. It is also guaranteed that any such proof will be extraordinarily difficult. For anyone seeking to make a career choice, I would suggest that you have a far better chance of becoming a winning Super Bowl quarterback than you do of proving the Riemann Conjecture.

Mathematics, however, is much easier on the knees.

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