ANDREW WILES

July 13, 2001

It?s my great pleasure to welcome the Olympiad contestants, their parents, organizers and others to the closing ceremony of this Olympiad. I'm going to address myself primarily to the contestants: the aspiring young mathematicians in what for some of you at least may be your graduation from high school mathematics. Unlike a traditional graduation, perhaps many of you will have no clear idea of what awaits you in the outside world of professional mathematics. However before I talk of the future, let me first congratulate you all. Some have arrived here by overcoming immense personal difficulties, others have arrived here overcoming only immense mathematical difficulties, but all of you have shown great talent and a real capacity for tremendous hard work.

Let me then welcome you not only to this event but also to the greater world of mathematics in what many of us believe is now a golden age. However let me also warn you — whatever the route you have taken so far, the real challenges of mathematics are still before you. I hope to give you a glimpse of this. What then distinguishes the mathematics we professional mathematicians do from the mathematical problems you have faced in the last week? The two principal differences I believe are of scale and novelty. First of scale: in a mathematics contest such as the one you have just entered, you are competing against time and against each other. While there have been periods, notably in the thirteenth, fourteenth and fifteenth centuries when mathematicians would engage in timed duels with each other, nowadays this is not the custom. In fact time is very much on your side. However the transition from a sprint to a marathon requires a new kind of stamina and a profoundly different test of character. We admire someone who can win a gold medal in five successive Olympics games not so much for the raw talent as for the strength of will and determination to pursue a goal over such a sustained period of time. Real mathematical theorems will require the same stamina whether you measure the effort in months or in years. You can forget the idea, if you ever had it, that all you require is a bit of natural genius and that then you can wait for inspiration to strike. There is simply no substitute for hard work and perseverance.

The second principal difference is one of novelty. It often comes as a surprise to non-mathematicians that we do place such a strong emphasis on originality. One common reaction is that mathematics has all been known for years; all we do is to perform calculations. This could not be further from the truth. Firstly there are many problems, old and new, that have not been solved. Secondly the creation of new mathematics enables us to dramatically simplify and clarify the old results. Calculus is now routinely taught in high school; however, in the seventeenth century it could not have been understood except by a very, very few eminent mathematicians in the world. This is not because people are smarter now, but because our language is clearer. The language of mathematics has been purified by the many new uses to which it has been put. Let me stress that creating new mathematics is a quite different occupation from solving problems in a contest. Why is this? Because you don't know for sure what you are trying to prove or indeed whether it is true. It is amazing how much difference it makes to know that a problem can be solved, and even if you believe as I do that most reasonably asked problems — mathematical problems at least — can be solved, it may be that you live in the wrong century. Some problems solved in the twentieth century could not conceivably have been solved in any previous century — simply too much new mathematics had to be developed first. What do you need to deal with this? You need the wisdom to choose problems that can be solved and the faith to trust in that judgment when it seems hopeless. I have been speaking perhaps as a pure mathematician, but mathematics is of course also the language of science and the language of technology. The demands in these directions may perhaps seem different, but I suspect that the profound contributions in these areas require the same blend of talent, hard work, and perseverance.

I've talked now enough in the abstract. Let me talk about one of these unsolved problems. I'm going to talk about a problem that is at least a thousand years old perhaps more. It is a part of one of the seven problems selected by the Clay Mathematics Institute as its Millennium Prize Problems For each of these, as you heard before, there is a prize of one million dollars. But I am getting ahead of myself. Let me begin with the prehistory of this problem.

You will see here written in clear text a sequence — you see here in ancient cuneiform script mathematical statements that were written down nearly four thousand years ago. These are preserved on tablets in Columbia University?s library and they were deciphered this century. They describe part of a problem which many of you know, but before you recognize it perhaps I should give you the translation of what is written in these tablets.

These were ancient Babylonian solutions to the equations x² + y² = z². Perhaps the first is a familiar one, 3² + 4² = 5², but there are also many others reaching what seems the incredibly large 18,541. You may be surprised that these fit on the tablet, but the Babylonians wrote in base 60, and it?s much easier to write a large number in base 60! We don?t know to what uses the Babylonians put such results. It's always been assumed that the Babylonians only did mathematics for its uses. Why this is assumed I don't know, it may not be true. These solutions are of course related to what we now call Pythagoras' theorem.

We think it most unlikely that the Babylonians had proved Pythagoras' theorem. On the other hand there are many people who think it's unlikely that Pythagoras proved Pythagoras' theorem, so perhaps we should not hold that against them. Pythagoras' theorem says of course that the square on the hypotenuse, c² in this picture, is equal to a² + b², the sum of the squares on the two shorter sides. I believe the first proof is found in Euclid. We do not know for sure whether Pythagoras solved this, but what we do know is that the Greek period marked a crucial transition from examining special cases and algorithms to the axiomatic method and to the unifying concept of proof. This idea of proving theorems rather than doing calculations, as I have said, is the bread and butter of modern mathematics, even though some rather provocative observers believe that this era is now ending and that we will be replaced by machines.

Well what does this have to do with the problem I was going to explain? The problem I am going to explain which, as I said, can be viewed as part of one of the Clay Mathematics Institute problems, is the following: suppose we take such a right angle triangle. We know from Pythagoras? theorem that a² + b² = c². The question is, if you pick an integer n, does there exist such a right angle triangle where the lengths of the sides are all rational numbers and the area is n? This problem has a long history. We can date it back at least to Arab manuscripts of the tenth century. Perhaps it goes back to the Greeks — we don't know. Fibonacci was challenged to give the answer for n = 5 in the court of Frederick II, and he achieved it. He then gave an incorrect proof for the case n = 1. And it was in fact Fermat who wrote down the first correct proof for the case n = 1. That is, he showed that there is no right angle triangle with area 1 whose sides are rational. Fermat's margin had room for this one! In fact it was probably the only complete proof he really wrote out, though there were one or two sketches of others. Well in principle it seems at least easy to write down triangles with given area; for example, you can write down a 3-4-5 triangle. The area being half the base times the height, is 6. So that 6 is the area of a right angle triangle. These for historical reasons have become known as congruent numbers, the ones that are the areas of right angle triangles. The problem is, how do you decide in general whether an integer n is the area of a right angle triangle with rational sides? There is no known algorithm. However, in the last 20 years, a truly remarkable conjecture has been developed which will give an answer to this question.

So for the question when does n occur as the area of a right angle triangle with rational sides, the conjectured answer for n odd and square-free is yes, if and only if the number of solutions to the equation 2x² + y² + 8z² = n is twice the number of solutions of 2x² + y² + 32z² = n. Although this problem has been around for a thousand years, it is not surprising that it took almost all those thousand years to find such a remarkable conjectural answer. This answer seems to come completely out of the blue, and although both the statement and the hypothetical answer are completely elementary, it seems that the relation between them is extremely deep and sophisticated. Even to make this guess required the development of much of twentieth century mathematics. Behind this problem lies a fantastic edifice that is now modern mathematics. Why is this? We don't know. Why do such simple sounding statements require such sophistication? We don't know. We believe that this is correct, and as I said, it is only one case, but it will be a crucial case, in one of the Clay Mathematics Institute problems.

Let me show you how it works. It's very simple then to make these calculations. To find the solutions of 2x² + y² + 8z² = n, for example take n = 11. There are twelve solutions and they are listed there. You take the solutions for 2x² + y² + 32z² = n, and again they are listed there. There are twelve solutions to the first and four to the second. Twelve is not twice times four, so there is no solution. If, as in the cases of 5 and 7, the number of solutions to the first equation is twice the number of solutions to the second, then there is a right angle triangle. So this gives an elementary and seemingly miraculous prescription for deciding the answer. Now I?m not saying that the answer to this riddle will change the world, it will not, but this problem expresses for some of us the magic of mathematics, that these strange interrelations between numbers can be studied and solved using the most sustained and wonderful arguments in modern mathematics. The modern mathematics we use to develop these problems and to solve these problems is itself extremely useful and has been applied in many contexts.

Even supposing you solve this problem, you still have the question: can you find the right angle triangle with that given area? It turns out that even if you could solve this problem, it is not obvious what the answer to that would be.

Here, it is perhaps hard to read, but this is actually the smallest solution for a right angle triangle with area 157 — smallest in the sense that the numerator and denominator are the smallest numbers possible. As you can see, this is not something that is found by trial and error, nor can it even be found simply using a computer. Behind this incredibly elementary problem that has been around for a thousand years, there is a very beautiful and very intricate mathematical theory which involves many of the modern branches: analysis, automorphic forms, algebra, most of the kinds of mathematics you've ever heard of. Just to pull out this one solution, knowing all that mathematics, in fact, is not too difficult.

Well I hope that this challenge and the other challenges you will understand once you know more mathematics will inspire you to follow mathematics as a career and to try your hand yourselves at one of these great problems. Thank you.