From Fermat to Wiles (And Those Other Guys)

The origin of Fermat’s Last Theorem by Fermat himself, which is clearly not written in English, found in a comment made in Diophantus’ Arithmetica.

In 1637, Pierre de Fermat made his infamous claim that the equation an+bn=cn, where a, b, and c are positive integers, cannot be satisfied for any integer n that is greater than 2. While this equation may look familiar from the Pythagorean Theorem, where a2+b2=c2, when those powers of 2’s are changed to 3’s, or any other value greater than 2 for that matter, there are no possible solutions with positive integers for a, b, and c. While Fermat claimed to have a proof that was too large to fit the margin, it is highly unlikely since the math needed to prove this theorem, or at least for this particular proof, as discovered in 1994 by Andrew Wiles, was not even known until centuries later. So how did this visually simple yet mathematically rigorous conjecture get proved over the course of 300 years?

Well, it begins with elliptic curves and modular forms. Elliptic Curves, as the MIT open course on such curves describes, “… are algebraic curves with the remarkable property that the set of points on the curve can be given the structure of an abelian group.” An abelian group is a group which commutes, that being ab=ba, where a and b are elements of said group. Anyway, elliptic curves, surprisingly enough, made their mathematical debut in Diophantus’ Arithmetica. These curves take the form y2=x3+ax+b, where a and b are rational numbers and the equation x3+ax+b has roots which are distinct.

The other half, in terms of main ideas, of Wiles’ proof of Fermat’s Last Theorem comes from modular forms. Modular forms’ beginnings come from Gauss, since seemingly everything in math stems from Gauss. In 1831 Gauss “gave a geometrical interpretation of some basic notions of number theory,” as a University of Michigan lecture puts it. From there came the ever so important modular forms. By definition, “a modular form of weight k and level N is an analytic function f from the complex upper half of the plane to the complex plane such that f(az+b/cz+d)=(cz+d)k*f(z) for all integers a, b, c, and d with ad-bc = 1 and c divisible by N.” This may sound like a whole lot of jargon (it sure does to me), so all that you need to know is that these modular forms are incredibly symmetric. As Harvard professor Barry Mazur puts it, “Modular forms are functions on the complex plane that are inordinately symmetric. They satisfy so many internal symmetries that their mere existence seem like accidents. But they do exist.”

An important idea to note about these seemingly accidental functions is that they do not seem to have anything in common or relating to elliptic curves. Well, until Yukata Taniyama had something to say about it at least. In 1956, Taniyama paved the way for his, as well as Goro Shimura’s, conjecture, which is creatively named the Taniyama-Shimura conjecture. This conjecture stated that rational elliptic curves are related to modular forms. Somehow these two seemingly two completely unrelated mathematical things are actually related to one another. While the 1956 Taniyama conjecture was not fully correct, as some errors were still in the conjecture, it was worked on by mathematicians such as Gerhard Frey, Ken Ribet, Jean-Pierre Serre, and Andre Weil over the course of four decades to get to the point where there was incredibly strong evidence pointing to its truth. It took until 2001 for the full modularity theorem to be proved, which as I stated previously is just elliptic curves under rational numbers relation to modular forms. The modularity theorem is the proof that finally proved the Taniyama-Shimura conjecture. Meanwhile for Wiles this connection was the key for his proof of Fermat’s Last Theorem. However, Wiles only needed the modularity theorem for semistable elliptic curves, which would be sufficient for him to prove Fermat’s Last Theorem.

In 1985, Gerhard Frey, a German mathematician, came up with the Frey curve which takes the form y2=x(x-al)(x+bl), or its equivalent y2=x(x-al)(x-cl). If this looks familiar in the least to Fermat’s Last Theorem it should, because this curve can be associated with a possible (hypothetical) solution of Fermat’s equation taking the form this time al+bl=cl, in the cases where l is prime. Back in the 1960’s Yves Hellegouarch originated the idea of associating Fermat’s equation with elliptic curves. Frey helped bring this to fruition with his Frey curves. If true, this would lead as a counterexample for Fermat’s Last Theorem, since a curve like this would not be modular, but if elliptic curves in reality are modular, Fermat’s Last Theorem would hold.

Jean-Pierre Serre, a French mathematician, believed Frey curves could not be modular, which like I just mentioned are a counterexample, letting the Taniyama-Shimura conjecture work with Fermat’s Last Theorem. However, Serre did not fully prove his idea, rather moving on to his own conjecture, Serre’s Conjecture, which would “imply Taniyama-Shimura conjecture,” which is separate from his original counterexample. However the missing part of his original proof was vital for Wiles’ proof and needed to be proved for any Wiles to progress. He needed to prove the epsilon conjecture to prove Fermat’s Last Theorem. While you may be completely lost in the math and terminology, an appreciation for the amount of effort not only Wiles but all the other mathematicians who paved the way for Wiles’ connections within his proof is one of the truly magical things about this proof. It feels like the Odyssey of math proofs; if Homer could create an epic based on a math proof this would certainly be it.

So at this point in time, the point at which the epsilon conjecture needed to be proved, University of California Berkley’s mathematics professor Ken Ribet made a huge stride in the metaphorical final turn of Wiles proof. Ribet proved the epsilon conjecture, proving that Fermat’s Last Theorem would follow from the Taniyama-Shimura conjecture. All that was left was to prove that the Taniyama-Shimura conjecture held up, and Wiles’ proof was next to done. Well, when I said Wiles’ proof was at its final turn, I may have jumped the gun (that’s two race analogies if you are counting!) The Taniyama-Shimura conjecture was seen as impossible to prove at the time, Ken Ribet even saying he was, “one of the vast majority who believed (it) was completely inaccessible.” Basically, the math just was not there. Sorry Wiles, better luck in the next life!

Wiles, however, was determined to finish this mammoth of a proof. While it seemed impossible, Wiles was able to get a second wind (that’s three, I can do this all day) and proved the Taniyama-Shimura conjecture. After writing 150 pages of the final proof, proving a theorem that was supposedly impossible for the time since the math was beyond the current knowledge, as well as countless (you can actually count them) years spent by other mathematicians creating the theorems and conjectures needed for the intermediate connections in the proof, Wiles proved that when an+bn=cn, with a, b, and c being positive integers, no solution can be found when n is greater than 2.