% History of Fermat's Last Theorem % by Andrew Granville % A version edited by Ian Katz appeared in The Guardian % (U.K.), on June 24 \magnification=\magstep1 \baselineskip=14pt It was announced yesterday that the most famous question in pure mathematics, Fermat's Last Theorem, has finally been answered. Englishman Andrew Wiles, of Princeton University, a Fellow of the Royal Society, told an academic audience at the Isaac Newton Institute in Cambridge, including leading authorities from around the world, that he had finally concluded one of the most controversial chapters in scientific history. The story of Fermat's Last Theorem can be traced back to ancient Greek times and stems from one of the best-known results in mathematics, Pythagoras' Theorem. That is, in any triangle which contains a right-angle, the square of the length of the hypoteneuse is equal to the sum of the squares of the lengths of the other two sides. (In mathematical notation, $z^2 = x^2 + y^2$, where $z$ is the length of the hypotenuse and $x$ and $y$ are the lengths of the other two sides.) This rule was extremely useful to the ancient Greeks because it allowed them to construct an accurate right-angle. The idea was simply to cut pieces of rope of lengths $x, y$ and $z$ feet, pull the ropes taut (holding the ends together) and, voil\a, a right angle. For this to be practical they needed $x, y$ and $z$ to each be whole numbers; and the example $5^2=3^2+4^2$ sufficed for their purposes. Other examples that they recorded included $13^2=5^2+12^2$ and even $8161^2=4961^2+6480^2$. One of the great intellectual masterpieces of the ancient Greek world was Diophantus' {\sl Arithmetic}. This work, available in Latin translation in the seveteenth century, was an important inspiration for the scientific renaissance of that period, read by Fermat, Descartes, Newton and others. Fermat, a jurist from Toulouse, studied mathematics as a hobby. He didn't formally publish his work but rather disseminated his ideas in letters, challenging others to match and/or admire his understanding. Fermat was evidently inspired by the {\sl Arithmetic} and made many notes in the margin of his copy. After Fermat's death, his son Samuel published these notes and amongst them was the following tantalizing sentence, beside the description of Pythagoras' Theorem: ...it is impossible for a cube to be written as a sum of two cubes or a fourth power to be written as a 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. I have a truly marvelous demonstration of this proposition which this margin is too narrow to contain''. In mathematical notation, one cannot find whole numbers $x,y,z$ and $n$, with $n$ bigger than $2$, for which $z^n = x^n + y^n$. Whether Fermat was being overly optimistic about his demonstration', we shall probably never know, but his argument has not been reproduced in the intervening three and a half centuries, despite no shortage of effort to do so. His remarkably simple, seductively simple, assertion is known as {\sl Fermat's Last Theorem}. Substantial prizes have been offered for its solution (the largest of which, the Wolfskehl prize, was rendered trivial by the German hyper-inflation of the twenties). The great mathematicians of history have been in two minds as to whether to work on it. Ernst Kummer, the German mathematician of the last century who did so much to establish modern algebra, wrote that Fermat's Last Theorem is more of a joke than a pinnacle of science'', yet his most important work originated in his failed attempts to prove it ! Fermat's Last Theorem is the question that mathematicians love to hate: it is really only one example of an equation, and then one that isn't so relevant in a general study of equations. On the other hand, there is no denying its charm and simplicity. As a question it goes in and out of favour, sometimes being dismissed as irrelevant' and unimportant', at other times hailed as the most interesting equation around. Many people have tried to give an unassailable proof' that Fermat's Last Theorem is true. It is a favourite of professional and amateurs alike: and, every few years, the newspapers report yet another purported proof, which is subsequently found to be lacking in some way. So why is Professor Wiles's attempt different ? And why are there many of us, who are usually extremely skeptical of such claims, prepared to believe that Wiles could have succeeded where so many others have failed ? To understand, it helps to have some perspective of the recent history of the problem: \ Up until the last decade the question seemed unassailable. To be sure, many interesting approaches have been proposed, persuading us of the truth of Fermat's Last Theorem, but without providing a proof that can be checked in every case. One such proof' is known to work for every exponent $n$ up to four million: that is, for every number $n$ up to four million there are no whole numbers $x, y$ and $z$ for which $z^n=x^n+y^n$ has solutions. However this method of proof involves an individual computer verification for each $n$, and so will not give the result for {\bf all} exponents $n$. In 1983, a young German mathematician, Gerd Faltings, proved a result far, far beyond what anyone had thought provable. Instead of looking only at the equations $z^n=x^n+y^n$ in isolation, he studied a very wide class of equations, the so-called {\sl curves}', and discovered exactly when they could and could not have infinitely many solutions in whole numbers. In particular, Faltings almost proved Fermat's Last Theorem; he showed that for any given $n>2$ there are only a few, that is finitely many, whole number solutions to $z^n=x^n+y^n$ (not allowing scaling'): however, it seems unlikely that his methods can be modified to actually show that there are {\sl no} solutions. Faltings' work revolutionized the way that people thought about solving equations, leading to the award of a Fields' Medal, the mathematical equivalent of the Nobel Prize. Wiles' approach to Fermat's Last Theorem comes, however, from a somewhat different direction, with rather different types of ideas. It all began in 1955, with a question posed by the Japanese mathematician Yutaka Taniyama: \ Could one explain the properties of {\sl elliptic curves}, equations of the form $y^2=x^3+ax+b$ with $a$ and $b$ given whole numbers, in terms of a few well-chosen curves. That is, is there some very special class of equations that in some way encapsulate everything there is to know about our elliptic curves ? Taniyama was fairly specific about these very special curves (the so-called {\sl modular curves}) and in 1968, Andr\'e Weil, brother of the philosopher Simone Weil, and himself one of the leading mathematicians of the century, made explicit which modular curve should describe which elliptic curve. In 1971 the first significant proven evidence in favour of this abstract understanding of equations was given by Goro Shimura, a Japanese mathematician at Princeton University, who showed that it works for a very special class of equations. This somewhat esoteric proposed approach to understanding elliptic curves, is now known as the Shimura-Taniyama-Weil conjecture. There the matter stood until 1986 when another young German researcher, this time Gerhard Frey from Saarbr\"ucken, made the most surprising and innovative link between this very abstract conjecture and Fermat's Last Theorem. What he realized was that if $c^n=a^n+b^n$ then it seemed unlikely that one could understand the equation $y^2=x(x-a^n)(x+b^n)$ in the way proposed by Taniyama. It took deep and difficult reasoning by Jean-Pierre Serre in Paris, and Ken Ribet in Berkeley, California, to strengthen Frey's original concept to the point that a counterexample to Fermat's Last Theorem would directly contradict the Shimura-Taniyama-Weil conjecture. This is the point where Wiles enters the picture. Wiles, already established as one of the deepest pure mathematicians of his generation, has drawn together a vast array of techniques to attack this question. Motivated by extraordinary new methods of Victor Kolyvagin of the Sieklov Institute in Moscow, and Barry Mazur of Harvard Univeristy, Wiles has succeeded in establishing the Shimura-Taniyama-Weil conjecture for an important class of examples, including those relevant to proving Fermat's Last Theorem. His work can be viewed as a blend of arithmetic and geometry, and has its origins way back in Diophantus' Arithmetic. However he employs the latest ideas from a score of different fields, from the theories of $L$-functions, group schemes, crystalline cohomology, Galois representations, modular forms, deformation theory, Gorenstein rings, Euler systems and many others. He uses, in an essential way, concepts due to mathematicians from Britain, France, Germany, Italy, Japan, the United States, Canada, Russia and Colombia; the culmination of work by many people around the world thinking about very different questions. Few suspected that their work might have this kind of application; and there are perhaps no more than half-a-dozen people in the world who are capable of fully understanding all the details of what Wiles has done. Given the enormous complexity of this work it will take some time to be certain that every detail Wiles relies upon, from such an array of areas, is correct. However, leading experts have now examined the essential ideas and there can be no question that his work is a profound contribution which sheds light on many important questions in pure mathematics. It is a tour de force, and will stand as one of the scientific achievements of the century. His work is not to be seen in isolation, but rather as the culmination of much recent thinking in many directions. There can be little doubt that over the next few years, mathematicians will shorten and simplify Wiles' proof, so that it will be accessible to a wider audience. (Wiles' proof, starting from scratch, would surely be over a thousand pages long). The story of this important discovery is a tribute to the deeper and more abtruse levels of abstract understanding that mathematicians have long claimed as essential. Many of us, while hailing Wiles' magnificent achievement, yearn for Fermat to have been correct, and for the truly marvellous, and presumably comparatively straightforward, proof to be recovered. \end