V.10 Fermat’s Last Theorem

Many people, even if they are not mathematicians, are aware of the existence of Pythagorean triples: that is, triples of positive integers (x, y, z) such that x² + y² = z². These give us examples of right-angled triangles with integer side lengths, of which the best known is the “(3, 4, 5) triangle.” For any two integers m and n, we have that (m²–n²)² + (2mn)² = (m² + n²)², which gives us an infinite supply of Pythagorean triples, and in fact every Pythagorean triple is a multiple of a triple of this form.

FERMAT [VI.12] asked the very natural question of whether similar triples existed for higher powers: that is, could there be a solution in positive integers of the equation xⁿ + yⁿ = zⁿ for some power n ≥ 3? For instance, is it possible to express a cube as a sum of two other cubes? Or rather, Fermat famously claimed that it was not possible, and that he had a proof that space did not permit him to write down. Over the next three and a half centuries, this problem became the most famous unsolved problem in mathematics. Given the amount of effort that went into it, one can be virtually certain that Fermat did not in fact have a proof: the problem appears to be irreducibly difficult, and solvable only by techniques that were developed much later than Fermat.

The fact that Fermat’s question was an easy one to think of does not on its own guarantee that it is interesting. Indeed, in 1816 GAUSS [VI.26] wrote in a letter that he found it too isolated a problem to interest him. At the time, that was a reasonable remark: it is often extremely hard to determine whether a given Diophantine equation has a solution, and it is therefore easy to come up with hard problems of a similar nature to Fermat’s last theorem. However, Fermat’s last theorem has turned out to be exceptional in ways that even Gauss could not have been expected to foresee, and nobody would now describe it as “isolated.”

By the time of Gauss’s remark, the problem had been solved for n = 3 (by EULER [VI.19]) and n = 4 (by Fermat; this is the easiest case). The first serious connection between Fermat’s last theorem and more general mathematical concerns came with the work of KUMMER [VI.40] in the middle of the nineteenth century. An important observation that had been made by Euler is that it can be fruitful to study Fermat’s last theorem in larger RINGS [III.81 §1], since these, if appropriately chosen, allow one to factorize the polynomial zⁿ - yⁿ. Indeed, if we write 1, ζ, ζ², . . ., ζ^n–1 for the nth roots of 1, then we can factorize it as

Therefore, if xⁿ + yⁿ = zⁿ then we have two rather different-looking factorizations of xⁿ inside the ring generated by 1 and ζ (namely the factorization in (1) above, and xxx · · · x), and it is reasonable to hope that this information might be exploited. However, there is a serious problem: the ring generated by 1 and ζ does not enjoy the UNIQUE FACTORIZATION PROPERTY [IV.1 §§4–8], so one’s sense of being close to a contradiction when faced with these two factorizations is not well-founded. Kummer, in connection with the search for HIGHER RECIPROCITY LAWS [V.28], had met this difficulty and had defined the notion of an IDEAL [III.81 §2]: very roughly, if you enlarge a ring by adding in Kummer’s “ideal numbers,” then unique factorization is restored. Using these concepts, Kummer was able to prove Fermat’s last theorem for every prime number p that was not a factor of the CLASS NUMBER [IV.1 §7] of the corresponding ring. He called such primes regular. This connected Fermat’s last theorem with ideas that have belonged to the mainstream of ALGEBRAIC NUMBER THEORY [IV.1] ever since. However, it did not solve the problem, since there are infinitely many irregular primes (though this was not known in Kummer’s day).

It turned out that more complicated ideas could be used for individual irregular primes, and eventually an algorithm was developed that could check for any given n whether Fermat’s last theorem was true for that n. By the late twentieth century, the theorem had been verified for all exponents up to 4 000 000. However, a general proof came from a very different direction.

The story of the eventual proof by Andrew Wiles has been told many times, so we shall be very brief about it here. Wiles did not study Fermat’s last theorem directly, but instead solved an important special case of the Shimura–Taniyama–Weil conjecture, which connects ELLIPTIC CURVES [III.21] and MODULAR FORMS [III.59]. The first hint that elliptic curves might be relevant came when Yves Hellegouarch noticed that the elliptic curve y² = x(x - a^p)(x - b^p) would have rather unusual properties if a^p + b^p was also a pth power. Gerhard Frey realized that such a curve might be so unusual that it would contradict the Shimura–Taniyama–Weil conjecture. Jean-Pierre Serre came up with a precise statement (the “epsilon conjecture”) that would imply this, and Ken Ribet proved Serre’s conjecture, thus establishing that Fermat’s last theorem was a consequence of the Shimura–Taniyama–Weil conjecture. Wiles suddenly became very interested indeed, and after seven years of intensive and almost secret work he announced a solution to a case of the Shimura–Taniyama–Weil conjecture that was sufficient to prove Fermat’s last theorem. It then emerged that Wiles’s proof contained a serious mistake, but with the help of Richard Taylor he managed to find an alternative and correct argument for that portion of the proof.

The Shimura–Taniyama–Weil conjecture asserts that “all elliptic curves are modular.” We finish by giving a rough idea of what this means. (A few more details can be found in ARITHMETIC GEOMETRY [IV.5].) Associated with any elliptic curve E is a sequence of numbers a_n (E), one for each positive integer n. For each prime p, a_p(E) is related to the number of points on the elliptic curve (mod p); it is easy to derive from these values the values of a_n(E) for composite n. Modular forms are HOLOMORPHIC FUNCTIONS [I.3 §5.6] with certain periodicity properties defined on the upper half-plane; associated with each modular form f is a FOURIER SERIES [III.27] that takes the form

f(q) = a₁(f)q + a₂(f)q² + a₃(f)q³ + · · · .

Let us call an elliptic curve E modular if there is a modular form f such that a_p (E) = a_p (f) for all but finitely many primes p. If you are presented with an elliptic curve, it is not at all clear how to set about finding a modular form associated with it in this way. However, it always seemed to be possible, even if the phenomenon was a mysterious one. For instance, if E is the elliptic curve y² + y = x³ – x² – 10x – 20, then there is a modular form f such that a_p (E) = a_p (f) for every prime p apart from 11. This modular form is the unique complex function (up to scaling) that satisfies a certain periodicity property with respect to the group Γ_o(11), which consists of all matrices () such that a, b, c, and d are integers, c is a multiple of 11, and the DETERMINANT [III.15] ad – bc is 1. It is far from obvious that a definition of this type should have anything to do with elliptic curves.

Wiles proved that all “semistable” elliptic curves are modular, not by showing how to associate a modular form with each such elliptic curve, but by using a subtle counting argument that guaranteed that the modular form had to exist. The full conjecture was proved a few years later, by Christophe Breuil, Brian Conrad, Fred Diamond, and Richard Taylor, which put the icing on the cake of one of the most celebrated mathematical achievements of all time.