Fermat’s Last Theorem

In June of 1993, the British mathematician Andrew Wiles (b. 1953) delivered a series of lectures on “Modular Forms, Elliptic Curves, and Galois Representations.” Though the audience was not large—the topic required a large amount of background knowledge—it had grown beyond expectations. It was rumored that Wiles would announce something out of the ordinary. And as he began a long discussion of a specific case of the Taniyama-Shimura conjecture, excitement increased. Finally, he reached the end, drawing an arrow on the blackboard which pointed to the letters FLT. What he had just proven entailed Fermat’s Last Theorem.

This was not actually a theorem, but rather a famously unsolved conjecture by the French lawyer and mathematician Pierre Fermat (1607-1665). Its origin was a tantalizing marginal annotation made in his copy of Arithmetica, a 3rd-century Greek textbook on Diophantine equations:

To divide a cube, however, into two cubes, or a fourth power into two fourth powers, or generally any power beyond a square into two of the same, is not possible; I worked out a truly marvelous proof of this thing. The thinness of the margin would not contain it.

In more modern terminology, the conjecture is that aⁿ+bⁿ≠cⁿ  for all n>2, assuming integer solutions and disallowing trivial cases such as a=c, b=0. The case for n=2, of course, is that of Pythagorean triples, discussed on the page next to the annotation.

Fermat made many other insufficiently-proven mathematical observations throughout his papers—some in other margins of that same book—and practically all of them turned out to be true. So why not this last one as well? It certainly looked simple, and specific values of n would be a nice place to begin the quest. In fact, only prime numbers and the number four need be examined, because a^mn+b^mn, with n prime, simplifies to (a^m)ⁿ+(b^m)ⁿ. Such specific proofs, however, proved slow in coming; the first was that of the famous Leonard Euler (1707-1783) for n=3.

Euler showed that any set of solutions a, b, c would imply the existence of a smaller set, creating an impossible infinite regress. This technique, called the method of descent, had originated with Fermat himself; as it can be used to construct a faulty proof of the full theorem, it may well have been that “truly marvelous proof.” It was becoming clear that 17th-century mathematics wasn’t enough for a real proof, if it did exist. (Fermat never posed the question in his ample correspondence, so he likely realized this himself at some point.) But Fermat’s Last Theorem retained its fascination, both to reward-seeking amateurs versed only in the basic algebra of its statement and to serious researchers looking for connections in increasingly complex fields.

The first mathematician to obtain significant general results was Marie-Sophie Germain (1776-1831), who investigated the “first case” of Fermat’s Last Theorem. She found that this case, which restricts possibilities for a, b, and c to numbers not divisible by the exponent, would be true for a prime exponent p if one could find another prime q satisfying two conditions:

1.     If a, b, and c exist such that a^p+b^p+c^p is a multiple of q, then either a, b, or c is a multiple of q.
2.     There exists no such integer k such that k^p-p is a multiple of q.

Germain managed to find q for all p less than 100; many expressions exist, such as 2p+1, which, when prime, are always acceptable values of q. She failed, however, in her aims to prove that an infinite number of possible primes q exist for every prime p. This would have proven the full case of Fermat’s Last Theorem, but Germain herself later found specific values of p for which only a finite number of q are valid. Nonetheless, what she did prove, known as Sophie Germain’s Theorem, inspired much more advanced research into the first case.

False proofs remained extremely common into the 20th century. So did legitimate proofs for specific exponents, which had always been heavily algebraic, though more and more complicated with larger n. This was one of the things that took off with computers, and by the 1990s the theorem was verified for all n<4,000,000. But as the techniques involved were inapplicable to the general case, none of these special cases brought a proof of that any nearer. A counterexample, however large, could exist somewhere. The real progress came from quite a different area of math.

In the 1950s, two Japanese mathematicians—Yutaka Taniyama (1927-1958) and Goro Shimura (1930-2019)—conjectured that every elliptic curve corresponded to a modular form.1 This seemed far removed from Fermat’s simply-stated margin note until 1982, when Gerhard Frey (b. 1944) noticed that the elliptic curve y²+x(x-aⁿ)(x+bⁿ) acts very odd if one assumes that aⁿ and bⁿ sum to some cⁿ. Four years later, Kenneth Ribet (b. 1948) proved that such a curve, if in existence, would not have a modular form.

Andrew Wiles, then a professor at Princeton University, was “electrified” to learn of Ribet’s proof. The beauty of Fermat’s Last Theorem had captured him at a young age, even prompting some very inadequate proof attempts. Now the Taniyama-Shimura conjecture was the only piece missing from a true proof of the theorem by contradiction. Though only a specific case of the conjecture was needed, this was unquestionably a big piece, which many mathematicians thought unprovable with existing techniques. But Wiles threw himself into it.

Seven years later, the Cambridge lecture met much applause and media coverage. It wasn’t, however, definitive, as anyone familiar with academia knew. Wiles’s proof, over a hundred pages long, had to be checked and rechecked after he submitted it for publication. Then all this revealed an error: a subtle assumption regarding the computation of a particular group’s order. The rest of the proof was valid and extremely valuable, but the theorem everyone cared about did not follow.

Wiles promptly returned to his proof, frustrating outside speculation with his intense reclusiveness. After a year of hard work and one breakthrough in particular, he emerged with two new papers (one of them co-authored by Richard Taylor (b. 1962), a former student of his). And these did hold up to scrutiny. For more than three centuries, Fermat’s Last Theorem had motivated research in diverse mathematical fields; now it was finally true to its name.

Notes:

1. While elliptic curves have a topological definition, the equations writable in the form y²=x³+ax+b comprise almost all of them. As for modular forms, advanced group theory is needed to characterize them any further than a specific type of function.                         

Image credit:

https://commons.wikimedia.org/wiki/File:Pierre_de_Fermat.jpg

https://www.joh.cam.ac.uk/library/special_collections/early_books/fermat.htm

https://www.ox.ac.uk/news/2016-05-24-new-era-number-theory-sir-andrew-wiles-receives-abel-prize-mathematics

Sources:

https://www.wikipedia.org/

https://mathshistory.st-andrews.ac.uk/

https://mathworld.wolfram.com/

Kolata, Gina (24 June 1993). “At Last, Shout of ‘Eureka’ In Age-Old Math Mystery”. The New York Times.

Kolata, Gina (28 June 1994). “A Year Later, Snag Persists in Math Proof”. The New York Times.

Mahoney, Michael Sean. The Mathematical Career of Pierre de Fermat (1601-1665). Princeton: Princeton University Press, 1973.

Ribenboim, Paulo. Fermat’s Last Theorem for Amateurs. New York: Springer-Verlag, 1999.

Stewart, Ian. Significant Figures. New York: Hatchette Book Group, 2017.