Welcome to one of the most fascinating areas of mathematics. There’s a fair amount of work involved in understanding even approximately how the recent proof of this theorem was done, but if you like mathematics, you should find it very rewarding.
If you have ever read about number theory you probably know that (the so-called) Fermat’s Last Theorem has been one of the great unsolved problems of the field for three hundred and fifty years. You may also know that a solution of the problem was claimed very recently - in 1993. And, after a few tense months of trying to overcome a difficulty that was noticed in the original proof, experts in the field now believe that the problem really is solved.
In this report, we’re going to present an overview of some of the mathematics that has either been developed over the years to try to solve the problem (directly or indirectly) or else which has been found to be relevant. The emphasis here will be on the ”big picture” rather than technical details. (Of course, until you begin to see the big picture, many things may look like just technical details.)
We will see that this encompasses an astonishingly large part of the whole of ”pure” mathematics. In some sense, this demonstrates just how ”unified” as a science mathematics really is. And this fact, rather than any intrinsic utility of a solution to the problem itself, is why so many mathematicians have worked on it over the years and have treated it as such an important problem.
The statement of Fermat’s Last Theorem (FLT for short) is about as simple as any mathematical proposition could be:
The equation has no solution for non-zero integers , , and if is an integer greater than .
If you have heard about FLT at all, you probably know a little of the history as well, so we won’t go into its ”social” history here. If you’d like to know a little more or to refresh your memory, there are several online references for this:
If you really want to explore, you can start with this page of links for Fermat’s Last Theorem
How can something like FLT be proved? Since it is a statement about the non-existence of something, the proof has to be somewhat indirect. Of course, if one could actually find a solution for some set of numbers, that would disprove the theorem and solve the problem. But we want a proof that FLT is true. The ”easiest” way to show that something doesn’t exist is to show that the supposed existence would lead to a contradiction.
At the highest level, the proof is extremely simple to understand, since it follows from just two theorems:
is semistable but not modular.
And
However, both of these theorems are very difficult themselves, and both have been proven only in the last 10 years. But given that both are now known, it follows that, in order to avoid a contradiction, there cannot be any solution to the Fermat equation.
Don’t worry too much now about the terminology used in these theorems. The purpose of this report is to explain some of the terms and many related concepts - and in the process give a bird’s eye view of a vast amount of mathematical terrain.
Theorem A is obviously rather special in that it applies only if the Fermat equation has a solution. (And since we now know this isn’t the case, the theorem has no further use.) It was first conjectured around 1982 by Gerhard Frey, and finally proved in 1986 by Ken Ribet, with help along the way from Jean-Pierre Serre.
Theorem B is even harder still, and it is the theorem of which Andrew Wiles first claimed a proof in 1993, thus proving FLT as well. Although problems were found in Wiles’ original proof, he managed to nail it down a year later, with help from Richard Taylor.
Actually, Theorem B was conjectured earlier (in a special form) by Yutaka Taniyama around 1955, and increasingly more general forms since then by Goro Shimura and Andre Weil. It is a special case of what is now known as the Taniyama-Shimura Conjecture (which dispenses with the technical semistable requirement). And the latter conjecture is a special case of much more general conjectures that are part of what is known as the Langlands Program, after Robert Langlands.
Theorem B certainly seems, to one unfamiliar with the territory, to be quite technical and abstruse. However, on closer examination, it can be seen to be both surprising and beautiful. The reason is that it concerns two apparently quite different sorts of mathematical objects - elliptic curves and modular forms. Each of these is relatively simple and has been studied intensively for over 100 years. Along the way some very surprising parallels have been observed in the theory of each (which we will discuss). And the theorem states that the parallels are in fact the result of a fundamental underlying connection between the two.
Wiles and Taylor proved Theorem B only with the semistable restriction given here, but many experts believe that much more general versions may be true. This is a very popular area of active research at present, and a number of the experts are hard at work trying to prove generalizations.
We’re now going to give a whirlwind tour of number theory and related mathematical fields that are relevant to FLT and the concepts that have turned out to be fundamental to its proof. There will be many terms tossed out rather casually, and unless you have done graduate work in mathematics many will probably be unfamiliar at first. Don’t let that dismay you, though - we intend to provide explanations and hypertext links to begin fleshing out some of the concepts and interrelationships.
Of course, any reasonably complete understanding is attainable only by dedicated study of graduate level texts and (eventually) research papers. But, we think, it is possible to learn your way around the ideas enough to orient yourself and to see how things fit together. How far you want to go beyond that is up to you.
FLT is a statement in number theory. The earliest attempts to prove it, by founders of number theory such as Euler, Dirichlet, and Legendre, usually involved only ”elementary” techniques - that is, arguments which (though often very clever and creative) can be understood by anyone who knows what is now high school algebra.
Matters suddenly took a more profound turn when Kummer realized that necessary assumptions about unique factorization of numbers into primes that hold for ordinary integers fail for the generalized integers of an algebraic number field. (An algebraic number field is a finite ”extension” of the ordinary rational numbers to include the solutions of specific polynomial equations.) To solve this problem, Kummer invented a new kind of ”ideal” numbers where unique factorization still occurs. Several decades of refinement of Kummer’s ideals led directly to such ideas of modern algebra as rings, and then to modern algebraic number theory as we know it.
Despite the great power and importance of Kummer’s ideal theory, and the subtlety and sophistication of subsequent developments such as class field theory, attempts to prove FLT by purely algebraic methods have always fallen short.
But something else rather surprising happened. Bernard Riemann was one of the greatest mathematicians of the 19th century, perhaps best known for putting integral calculus on a rigorous footing (with the Riemann integral). But he did a lot more that’s quite relevant to number theory and FLT as well.
In the 1850s Riemann investigated the properties of a certain complex function called the zeta function, which had been of interest much earlier to people like Euler and Dirichlet. The zeta function is perhaps the simplest of a class of functions defined by a series expansion named after Dirichlet. The analytic behavior of this function, in particular the location of its zeros and poles, turned out to have a profound connection with the distribution of prime numbers. Knowledge of the zeta function eventually allowed Hadamard to prove the ”prime number theorem”, which gives an asymptotic formula for the number of primes there are less than any given bound. A stronger and still unproven conjecture about the zeta function, the Riemann Hypothesis (which says that the only zeros of the zeta function in the strip lie on the line ), implies much more precise information about the distribution of primes.
Over the years, other mathematicians have invented and investigated generalizations of the zeta functions and Dirichlet series which turn out to be as intimately involved with generalizations of the ordinary rational numbers as the zeta function is with the rational numbers themselves. For instance, there are zeta functions of finite algebraic extensions of the rationals, and similar functions called L-functions that express facts about the Galois group of the extension field. There are also zeta and L-functions of elliptic curves and of finite fields. There are even p-adic analogues of zeta and L functions, defined over p-adic fields.
Various analogues of zeta and L-functions are used heavily in number theory and related areas. In particular, it is possible to formulate an equivalent of the Taniyama-Shimura conjecture as the assertion that for every elliptic curve there is a modular form which has the same associated L-function. This represents a very tantalizing and deep relationship of algebraic and analytic mathematical objects.
Riemann, in a relatively brief career, fertilized a large number of mathematical fields. As if what we’ve already mentioned weren’t enough for anyone, he also made absolutely fundamental contributions to complex analysis by his invention of the concept of Riemann surfaces. A Riemann surface is a generalizaton of the complex plane and a natural domain of definition of analytic functions. Riemann surfaces make it possible to define and study in a natural way a very interesting class of functions called elliptic functions, which were investigated by Weierstrass. These turn out to be very closely related to elliptic curves (i. e., the sort of curve involved in Theorems A and B). By looking at functions defined on a different Riemann surface from that of elliptic functions one can construct another type of functions known as modular functions. Theorem B and more general forms of the Taniyama-Shimura Conjecture can be viewed in yet another way to affirm that there is a very significant relationship between modular functions and elliptic curves. But even well before that, modular functions have been investigated for their many properties that imply quite elegant number theoretic results.
Incidentally, Riemann was also responsible for Riemannian geometry, i. e. the study of curves and surfaces by techniques of differential calculus. In fact, as his invention of Riemann surfaces suggests, Riemann contributed as much to geometry as to analysis. Indeed, he did a great deal to unify the two fields. Such concepts as tensor calculus and differential manifolds are a direct result of his work - and they became the essential tools of Einstein’s general relativity theory.
That, then, is a very brief overview of the mathematical cast of characters which play leading roles in the eventual resolution of Fermat’s theorem. There are various directions you can take from here. Each direction will often draw on concepts and facts that lie in one or more of the other directions, so you will have to be willing to wait until you’ve explored them all to get the best understanding of what’s going on. With that willingness to accept ideas which are only explained elsewhere, you can choose almost any path for the next step:
There is a lot of heavy-duty math in the following pages. That’s the whole point. There’s no use in pretending you will get much out of the discussion unless you’ve had at least a couple of college-level math courses. If you’ve had the courses but perhaps forgotten a little, that’s OK. There are reminders of the basic definitions and a glossary. An introduction to abstract algebra (groups, rings, fields) is almost essential. A course in linear algebra would be nice too. Introductory calculus will come in handy sometimes. A course in complex analysis would be a real plus, but if you haven’t had it, you can get by if you take a lot of basic results on faith.
Elliptic curves are relatively simple objects that helped inspire the field of algebraic geometry because of some very special properties.
A modular function is something like an elliptic function. Both are special cases of automorphic functions, which means they are invariant under certain group operations on their domains of definition. This introduces considerations of group theory and symmetry into the study of complex functions and Riemann surfaces. There turn out to be many parallels between the theory of elliptic curves and that of modular functions, which have deep consequences for both theories.
These Dirichlet series and their generalizations tie together number theoretic and analytic information in deep and mysterious ways.
Another kind of mathematical construct which can be made for both elliptic curves and modular forms. We look at Galois groups and their representations as matrices over various rings, including the p-adic numbers.
This is a sketch of the results of Ribet (proving Theorem A) and Wiles (proving Theorem B). Together they prove FLT. You can read this first if you just want the highest-level overview of the proof.