In a certainly well-deserved recognition, the Norwegian Academy of Science and Letters has awarded the 2016 Abel Prize to Andrew Wiles of the University of Oxford, who in 1995 published a proof of Fermat’s Last Theorem, that centuries-old, maddening conjecture that an + bn = cn has no nontrivial integer solutions except for n = 2.
Fermat’s Last Theorem was first conjectured in 1637 by Pierre de Fermat in 1637, in a cryptic annotated marginal note that Fermat wrote in his copy of Diophantus’ Arithmetica. For 358 years, the problem tantalized generations of mathematicians, who sought in vain for a valid proof. In the 1995 edition of the Guiness Book of World Records, it was cited as the world’s most difficult mathematical problem, in part because of the large number of unsuccessful proofs through the ages. Some of these proofs were foolish, but others helped build modern number theory.
In the mid-1970s, Wiles had trained under Cambridge University based Australian mathematician John Coates, who had recently returned to England from teaching at Stanford University. They studied the arithmetic of elliptic curves, using the various methods, including Iwasawa theory. (A personal note: Prior to working with Wiles, in 1972 John Coates taught one of the present authors (Bailey) a course in Algebra at Stanford.)
The proof of Fermat’s Last Theorem had its origin in a series of results in the 1980s by Gerhard Frey of the University of Duisburg-Essen, Jean-Pierre Serre of the Centre National de la Recherché Scientifique College de France, and Ken Ribet of the University of California, Berkeley. From their work, it became clear that Fermat’s Last Theorem might be proven as a consequence of a limited form of the Taniyama-Shimura-Weil conjecture, which is now known as the modularity theorem.
Wiles, who had been fascinated by Fermat’s Last Theorem since childhood, decided to pursue a proof. After working in secret for several years on the project, on 24 June 1993, he announced his result in a lecture at Cambridge University.
Alas, a few months later a flaw was uncovered in his proof. Finally, in 1995, a full corrected proof was published in Annals of Mathematics, with one of the two final papers co-authored with Richard Taylor. The proof has stood the test of time — 20 years later no flaw has been uncovered.
We add our congratulations to Wiles for his landmark achievement, and hope that his example will inspire many other mathematicians to pursue lines of research traditionally thought to be “too difficult.” A brief outline of Andrew Wiles’ proof is presented in the Wikipedia article on the topic. Additional information on both Wiles and the Abel Prize, modeled on the Nobel prize, is available at the Norwegian Academy’s website, and in well-written articles in New Scientist and Nature.