• Kenneth A Ribet

Theorem 1.2 x 3+y = uz3 has no solutions with x,y,z∈A, ua unit in A, xyz6= 0. This certainly implies (FLT) 3. Proof: By homogeneity, we may assume that x,y,zare rela-tively prime. Factoring x 3+y = uz3 gives (x+y)(x+ζy)(x+ζ2y) = uz3, where the gcd of any 2 factors on the left divides λ:= 1 −ζ. If each gcd is 1, then each factor is a cube up to a unit.

Fermat's Last Theorem was until recently the most famous unsolved problem in mathematics. In the mid-17th century Pierre de Fermat wrote that no value of n greater than 2 could satisfy the equation ' x n + y n = z n,' where n, x, y and z are all integers. He claimed that he had a simple proof of this theorem, but no record of it has ever been found. Ever since that time, countless professional and amateur mathematicians have tried to find a valid proof (and wondered whether Fermat really ever had one). Then in 1994, Andrew Wiles of Princeton University announced that he had discovered a proof while working on a more general problem in geometry.

Grundman, associate professor of mathematics at Byrn Mawr College, assesses the state of that proof: 'I think it's safe to say that, yes, mathematicians are now satisfied with the proof of Fermat's Last Theorem. Few, however, would refer to the proof as being Wiles's alone. The proof is the work of many people. Wiles made a significant contribution and was the one who pulled the work together into what he thought was a proof. Although his original attempt turned out to have an error in it, Wiles and his associate Richard Taylor were able to correct the problem, and so now there is what we believe to be a correct proof of Fermat's Last Theorem.

'The proof we now know required the development of an entire field of mathematics that was unknown in Fermat's time. The theorem itself is very easy to state and so may seem deceptively simple; you do not need to know a lot of mathematics to understand the problem. It turns out, however, that to the best of our knowledge, you do need to know a lot of mathematics in order to solve it. It is still an open question whether there may be a proof of Fermat's Last Theorem that involves only mathematics and methods that were known in Fermat's time.

We have no way of answering unless someone finds one.' Stevens in the mathematics department at Boston University expands on these thoughts: 'Yes, mathematicians are satisfied that Fermat's Last Theorem has been proved. Andrew Wiles's proof of the 'semistable modularity conjecture'-the key part of his proof-has been carefully checked and even simplified.

It was already known before Wiles's proof that Fermat's Last Theorem would be a consequence of the modularity conjecture, combining it with another big theorem due to Ken Ribet and using key ideas from Gerhard Frey and Jean-Pierre Serre. 'I would ask this second question the other way around. How did we get so lucky as to find a proof at all? The German polymath Karl Gauss summed up the attitudes of many pre-1985 professional mathematicians when in 1816 he wrote: 'I confess that Fermat's Last Theorem, as an isolated proposition, has very little interest for me, because I could easily lay down a multitude of such propositions, which one could neither prove nor dispose of.' Somehow we got lucky and managed to save Fermat's Last Theorem from its isolation by linking it to some important branches of modern mathematics, especially the theory of modular forms. Was this really just luck?

How many others of Gauss's 'multitude of propositions' can also be magically transformed and made accessible to the powerful tools of modern mathematics? Fermat's Last Theorem is just the beginning. There are many fascinating explorations still ahead of us!' And Fernando Q. Official files.

Gouva, chair of the department of mathematics and computer science at Colby College, offers some additional information: 'The full proof of Fermat's Last Theorem is contained in two papers, one by Andrew Wiles and one written jointly by Wiles and Richard Taylor, which together make up the whole of the May 1995 issue of the Annals of Mathematics, a journal published at Princeton University. Journal publication implies, of course, that the referees were satisfied that the paper was correct. 'In the summer of 1995, there was a large conference held at Boston University to go over the details of the proof. Specialists in each of the relevant areas gave talks explaining both the background and the content of the work of Wiles and Taylor.

After having subjected the proof to such close scrutiny, the mathematical community feels comfortable that it is correct. 'The second question is much harder to answer. It could very well be, of course, that the reason the theorem has taken so long to prove is that we have not been smart enough!

But that seems unlikely, seeing that so many brilliant mathematicians thought about it over the centuries. Why then was the proof so hard? 'Well, the first thing is that Fermat's Last Theorem is a very sweeping, general statement: for no exponent n greater than 2 is there a solution to the Fermat equation. It is much easier to attack the problem for a specific exponent. In one of his letters, for example, Fermat explains how to prove it for n = 4; Euler in the 18th century could produce a proof for the case of n = 3, and so on. In fact, just before Wiles's work, mathematicians had shown that there are no solutions to the theorem for numbers up to n = 4,000,000 or so.

That may seem like a lot of numbers, but of course, it doesn't even scratch the surface of a claim that talks about every exponent. 'The other problem is that Fermat's claim has always felt, well, marginal. It is hard to connect the Last Theorem to other parts of mathematics, which means that powerful mathematical ideas can't necessarily be applied to it. In fact, if one looks at the history of the theorem, one sees that the biggest advances in working toward a proof have arisen when some connection to other mathematics was found. For example, Polish mathematician Ernst Eduard Kummer's work in the mid-19th century arises from connecting the Last Theorem to the theory of cyclotomic fields. And Wiles is no exception: his proof grows out of work by Frey, Serre and Ribet that connects Fermat's statement with the theory of elliptic curves. Once that connection was established, and one knew that proving the Modularity Conjecture for elliptic curves would yield a proof of Fermat's Last Theorem, there was reason to be hopeful.

Wiles's work shows that such hope was justified.

Kenneth A Ribet

Wiles's proof of is a, by British mathematician, of a special case of the for. Together with, it provides a proof for. Both Fermat's Last Theorem and the modularity theorem were almost universally considered inaccessible to proof by contemporaneous mathematicians, seen as virtually impossible to prove using current knowledge.

Wiles first announced his proof on Wednesday 23 June 1993 at a lecture in entitled 'Elliptic Curves and Galois Representations.' However, in September 1993 the proof was found to contain an error. One year later, on Monday 19 September 1994, in what he would call 'the most important moment of his working life,' Wiles stumbled upon a revelation, 'so indescribably beautiful. So simple and so elegant,' that allowed him to correct the proof to the satisfaction of the mathematical community. The correct proof was published in 1995.

Wiles' proof uses many techniques from and, and has many ramifications in these branches of mathematics. It also uses standard constructions of modern algebraic geometry, such as the of and, and other 20th-century techniques not available to Fermat. Together, the two papers which contain the proof are 129 pages long, and constructing the proof consumed over seven years of Wiles's research time. Described the proof as one of the highest achievements of number theory, and called it the proof of the century.

Wiles' path to proving Fermat's Last Theorem, by way of proving the modularity theorem for the special case of, established powerful techniques and opened up entire new approaches to numerous other problems. For solving Fermat's Last Theorem, he was, and received other honours such as the 2016. When announcing that Wiles had won the Abel Prize, the described his achievement as a 'stunning proof.'