What Is Mathematics? An Elementary Approach to Ideas and Methods, 2nd Edition (1996)

§3. FERMAT’S LAST THEOREM

One of the most dramatic developments since Courant and Robbins wrote What Is Mathematics? was the 1994 proof of Fermat’s Last Theorem by Andrew Wiles of Princeton University. Recall that Fermat conjectured that the equation

(1) xⁿ + yⁿ = zⁿ

has no nonzero integer solutions when n ≥ 3. Wiles’s proof is highly technical and accessible only to experts. However, the general outline is comprehensible. The attack is highly indirect, and makes heavy use of the theory of “elliptic curves,” which are defined by Diophantine equations of the form

(2) y² = ax³ + bx² + cx + d

for rational numbers a, b, c, d. (The adjective “elliptic” derives from connections with so-called elliptic functions, and does not refer to the curve’s shape.) A great deal is known about such equations: they constitute one of the deepest and best understood areas of number theory.

Fermat’s equation (1) can be rewritten as (x/z)ⁿ + (y/zⁿ) = 1, so the point (X, Y) = (x/z, y/z) lies on the Fermat curve with equation

(3) Xⁿ + Yⁿ = 1.

Say that (X, Y) is a rational point if both X and Y are rational numbers. Then Fermat’s Last Theorem is equivalent to the assertion that no rational point can lie on the Fermat curve (3) when n ≥ 3. Between 1970 and 1975, Yves Hellegouarch investigated a curious connection between Fermat curves (3) and elliptic curves (2). Jean-Pierre Serre suggested trying the converse: to exploit properties of elliptic curves to prove results on Fermat’s Last Theorem. In 1985 Gerhard Frey made this suggestion precise by introducing what is now called the Frey elliptic curve associated with a presumptive solution of the Fermat equation. Suppose that there is a nontrivial solution Aⁿ + Bⁿ = Cⁿ of the Fermat equation, and form the elliptic curve

(4) y² = x(x + Aⁿ)(x – Bⁿ).

This is the Frey elliptic curve, and it exists if and only if Fermat’s Last Theorem is false. So in order to prove Fermat’s Last Theorem it is enough to prove that Frey’s curve (4) cannot exist. The way to do this is to follow the “indirect” method of proof (see p. 86): that is, to assume that it does exist and deduce a contradiction. This implies that the Frey curve does not exist after all, which implies that Fermat’s Last Theorem is true. Frey found strong evidence that his curve “ought not to exist” by proving that it has several extremely curious and unlikely sounding properties. In 1986 Kenneth Ribet pinned the probem down by proving that Frey’s curve cannot exist provided that a big unsolved problem in number theory, the Taniyama conjecture, is true. He thereby reduced one major unsolved problem, Fermat’s Last Theorem, to another major unsolved problem. This kind of reduction is often unhelpful, just replacing one hard problem by a harder one, but in this case it hit paydirt, because it provided a context in which to tackle Fermat’s Last Theorem.

The Taniyama conjecture is again technical, but it can be explained with reference to a special case. There is an intimate relationship between the “Pythagorean equation” a² + b² = c², the unit circle, and the trigonometric functions sin and cos. To find this relationship, observe that the Pythagorean equation can be rewritten in the form (a/c)² + (b/c)² = 1, which implies that the point (x, y) = (a/c, b/c) lies on the unit circle, whose equation is x² + y² = 1. It is well known that the trigonometric functions provide a simple way to represent the unit circle. Specifically, Pythagoras’s Theorem and the geometric definition of sin and cos imply that the equation

(5) cos² θ + sin² θ = 1

holds for any angle θ (see p. 277). If we set x = cos θ, y = sin θ, then (5) states that the point (x, y) lies on the unit circle. To sum up: solving the Pythagorean equation in integers is equivalent to finding an angle θ such that both cos θ and sin θ are rational numbers (equal respectively to a/cand b/c). Because the trigonometric functions have all sorts of pleasant properties, this idea is the basis of a really fruitful theory of the Pythagorean equation.

The Taniyama conjecture says that (in a rather technical setting) a similar kind of idea can be applied to any elliptic curve, but replacing sin and cos by more sophisticated “modular” functions. So problems about elliptic curves can be replaced by problems about modular functions, just as problems about the circle can be replaced by problems about trigonometric functions.

Wiles realized that Frey’s approach can be pushed through to a satisfactory conclusion without using the full force of the Taniyama conjecture. Instead, a particular case suffices, one that applies to a class of elliptic curves known as “semistable.” In a 100-page paper he marshalled enough powerful machinery to prove the semistable case of the Taniyama conjecture, leading to the following theorem. Suppose that M and N are distinct nonzero relatively prime integers such that MN(M — N) is divisible by 16. Then the elliptic curve y² = x(x + M)(x + N) can be parametrized by modular functions. Indeed the condition on divisibility by 16 implies that this curve is semistable, so the semistable Taniyama conjecture establishes the desired property.

We now apply Wiles’s theorem to Frey’s curve (4) by letting M = Aⁿ, N = –Bⁿ. Then M – N = Aⁿ + Bⁿ = Cⁿ, so MN(M – N) = –AⁿBⁿCⁿ, and we must show this is a multiple of 16. Now at least one of A, B, C must be even—for if A and B are both odd then Cⁿ is a sum of two odd numbers, hence even—which implies that C is even. We may further assume that n ≥ 5, because Euler long ago proved Fermat’s Last Theorem for n = 3. But since the fifth or higher power of an even number is divisible by 2⁵ = 32, the number –AⁿBⁿCⁿ is a multiple of 32, hence certainly a multiple of 16. Therefore Frey’s curve satisfies the hypothesis of Wiles’s theorem, implying that it can be parametrized by modular functions. However, Ribet’s proof that the Taniyama conjecture implies the nonexistence of Frey’s curve works by proving that the Frey curve cannot be parametrized by modular functions. This is a contradiction, so Fermat’s Last Theorem is true.

This proof is very indirect and requires sophisticated ideas. Moreover, some difficulties emerged concerning the first version of Wiles’s proof, which added to the sense of drama. He circulated a message by electronic mail to the mathematical community, acknowledging these difficulties but asserting his confidence that his methods would overcome them. Repairing the proof took longer than hoped, but on 26 October 1994 Karl Rubin circulated another message: “As most of you know, the argument described by Wiles… turned out to have a serious gap, namely the construction of an Euler system. After trying unsuccessfully to repair that construction, Wiles went back to a different approach, which he had tried earlier but abandoned in favour of the Euler system idea. He was then able to complete his proof.”