Sir Andrew Wiles (born 1953) is the British mathematician who, in 1994, proved Fermat’s Last Theorem — a 350-year-old open problem that had defeated every great mathematician since Pierre de Fermat first scribbled his famous margin note in 1637.
Life
Wiles was born in Cambridge, England, the son of a theology professor. He has said that at age ten, browsing a book in a local library, he came across the statement of Fermat’s Last Theorem — that has no positive integer solutions for — and resolved to solve it. He carried that quiet goal through his education and early career.
He earned his PhD at Cambridge in 1980 under John Coates, working on elliptic curves. He took a professorship at Princeton in 1982, where he remained for most of his career. He returned to Oxford as Royal Society Research Professor in 2011.
Fermat’s Last Theorem
In 1637, Pierre de Fermat wrote in the margin of his copy of Diophantus’s Arithmetica that he had found a “truly marvelous proof” that has no integer solutions for , but the margin was too small to contain it. No such proof was ever found among his papers. The conjecture resisted three centuries of attacks by every mathematician who tried.
The path through modularity
In 1986, Ken Ribet proved that a special case of the Taniyama-Shimura-Weil conjecture — a deep connection between elliptic curves and modular forms — would imply Fermat’s Last Theorem. Wiles learned of this result and decided to go after the conjecture itself.
He worked on the problem in near-secrecy for seven years, from 1986 to 1993. He told only his wife and one colleague (Nicholas Katz) what he was doing. During this time he deliberately reduced his other research activity, fearing that revealing interest in Fermat would make concentrated work impossible.
The announcement and the gap
In June 1993, Wiles gave a series of three lectures at the Isaac Newton Institute in Cambridge. At the end of the third, he announced he had proved Fermat’s Last Theorem. The news made international headlines.
Then, in September 1993, a gap was found in the proof during the refereeing process. A crucial step — bounding a specific cohomology group — didn’t quite work as Wiles had thought. For a year he and his former student Richard Taylor struggled to fix it. In September 1994, Wiles found a new approach that, combined with Taylor’s earlier technique, completed the proof.
The corrected proof was published in two papers in Annals of Mathematics in 1995: Wiles’s “Modular elliptic curves and Fermat’s Last Theorem” and a joint paper with Taylor, “Ring-theoretic properties of certain Hecke algebras.” Fermat’s 358-year-old conjecture was finally a theorem.
The broader result
What Wiles actually proved was the Taniyama-Shimura-Weil conjecture for semistable elliptic curves: that every semistable elliptic curve over the rationals is modular. This was a deep result in its own right, independent of Fermat. The full conjecture — that all elliptic curves over the rationals are modular — was proved by Breuil, Conrad, Diamond, and Taylor in 2001.
Recognition
Wiles was knighted in 2000. He received the Wolf Prize in 1995/6 (shared with Robert Langlands), the Shaw Prize in 2005, and in 2016 the Abel Prize — mathematics’ closest equivalent to the Nobel — “for his stunning proof of Fermat’s Last Theorem by way of the modularity conjecture for semistable elliptic curves, opening a new era in number theory.”
He did not receive the Fields Medal because he was 41 when the proof was completed, one year over the medal’s age limit. The IMU awarded him a special silver plaque at the 1998 Congress.
Legacy
Wiles’s proof is a monument of modern mathematics — a bridge between classical number theory and the deep machinery of modular forms, Galois representations, and elliptic curves. But just as important is what his achievement symbolized: that even the oldest, most resistant problems can yield to sustained effort, and that a mathematician can spend seven years underground on a single question and emerge with the answer.
“It was the most important moment of my working life,” he said. “Nothing I do again will mean as much.”
Known for
- Proof of Fermat's Last Theorem (1994)
- Modularity theorem (with Taylor)
- Abel Prize (2016)
Frequently asked
How long did Wiles work on Fermat's Last Theorem?
Wiles spent seven years working on it almost in secret, from 1986 to 1993. After announcing the proof in 1993, a gap was found. He and his former student Richard Taylor worked another year to fix it. The complete proof was published in 1995.
Why did Wiles work in secret?
Because the problem attracted so many cranks and amateurs that any mathematician announcing work on it would be harassed. And because Wiles believed that public attention would make concentrated work impossible. He told only one colleague about his efforts.
What did Wiles actually prove?
Wiles proved a special case of the Taniyama-Shimura-Weil conjecture: that semistable elliptic curves over the rationals are modular. This implies Fermat's Last Theorem via a connection found by Ken Ribet in 1986.