sourced story
1993-1995 CEPrimary source · 3 sourcesWell documented

Fermat's Margin Note Waits 358 Years for Andrew Wiles

A proof the margin was too small to contain finally arrives in 1994, mid-repair, after a year-long gap almost sank it

On the timeline · around 1993-1995 CE · Modern MathematicsModern MathematicsFermat's Margin Note Waits 358 Years for Andrew Wiles18751900192519501975

Quick facts

Fermat's original claim
c. 1630, in the margin of Diophantus's Arithmetica
Wiles's announcement
23 June 1993, Cambridge
Gap fixed
19 September 1994, with Richard Taylor
Published proof
Annals of Mathematics, 1995

What happened

Pierre de Fermat wrote in the margin of his copy of Diophantus's Arithmetica around 1630 that no three positive integers a, b, and c can satisfy a to the power n plus b to the power n equals c to the power n for any integer n greater than 2, adding, in his own words, that he had discovered a truly marvelous proof which this margin is too small to contain. No such proof by Fermat has ever been found, and the claim, known as Fermat's Last Theorem, went unproved for over 350 years. Andrew Wiles, born in 1953, announced a proof in a lecture at Cambridge's Isaac Newton Institute in June 1993, ending simply, I will stop here. By December 1993 a gap had emerged in the argument, centered on bounding a mathematical structure called the Selmer group, and Wiles spent nearly a year trying to fix it. On 19 September 1994, working with his former student Richard Taylor, Wiles found the fix using an approach tied to Barry Mazur's and Victor Flach's earlier methods, later describing the moment as the most important of his working life. The completed proof, Modular Elliptic Curves and Fermat's Last Theorem, was published in the Annals of Mathematics in 1995. Wiles received the Abel Prize in 2016, with the official citation praising his proof of Fermat's Last Theorem by way of the modularity conjecture for semistable elliptic curves as opening a new era in number theory.

Why it matters

Wiles's proof closed the single most famous open problem in mathematics, one that had outlasted every mathematician who attempted it for over three and a half centuries, and it did so not through elementary methods Fermat could plausibly have had access to but through 20th-century tools, elliptic curves and modular forms, that make historians doubt Fermat's original marginal claim was actually a valid proof at all. The episode is also a rare public demonstration of how modern mathematical proof actually works: a years-long process with an announced result, a discovered flaw, and a slow, collaborative repair, rather than a single flash of insight.

How we know

Wiles's completed proof was published and peer-reviewed in the Annals of Mathematics in 1995, and his own account of the 1993 gap and the September 1994 breakthrough, including his description of it as the most important moment of his working life, is recorded in his own retrospective statements collected by MacTutor and in the mathematical community's documented response to the announcement and correction.

Sources

See something wrong? . Corrections with a source get fixed fastest.

Part of a timelineHistory of Mathematics26 events · A number system built for taxes, a theorem older than the man it's named for, a proof too long for a margin, and an infinity too big to countView all →
Fermat's Margin Note Waits 358 Years for Andrew Wiles · History of Mathematics · SourcedStory