Some time around 1637, a lawyer in Toulouse sat reading his copy of the Arithmetica, a treatise by the ancient Greek mathematician Diophantus, and scribbled a note in the margin that would torment mathematics for three and a half centuries. Beside a problem about splitting one square number into two, Pierre de Fermat claimed that the equation xⁿ + yⁿ = zⁿ has no whole-number solutions for any power n greater than 2. Then came the most famous tease in mathematical history: “I have discovered a truly marvellous demonstration of this proposition, which this margin is too narrow to contain.” He never wrote it anywhere else. Fermat died in 1665, and the world only learned of the claim when his son Samuel published an edition of the Arithmetica in 1670 with his father’s marginal notes included.
Fermat was an amateur in the literal sense — a magistrate who did mathematics in the evenings — but one of the finest minds of the seventeenth century, and he had a habit of announcing results without proofs. Over the next two centuries every other claim of his was settled. This single stubborn survivor became known as Fermat’s Last Theorem: last because it was the last left standing, not the last he wrote. Almost no historian believes he possessed a general proof; the tools required did not exist for another 350 years. Most likely he had a genuine proof for the case n = 4 — which survives, built on his method of “infinite descent” — and briefly believed the idea would generalise.
Three centuries of near misses
The assault proceeded one exponent at a time. Leonhard Euler dealt with n = 3 in the eighteenth century. Sophie Germain, corresponding under the male pseudonym Monsieur LeBlanc because women were barred from serious mathematics, produced the first broad strategy in the early 1800s, with strong results for a whole class of primes now named after her. Dirichlet and Legendre settled n = 5 in 1825, Gabriel Lamé n = 7 in 1839, and in the mid-nineteenth century Ernst Kummer’s theory of “ideal numbers” swept up every “regular” prime at a stroke. In 1908 the German industrialist Paul Wolfskehl bequeathed 100,000 marks for the first proof, valid until 2007 — a prize that attracted sackfuls of crank submissions and, after hyperinflation, lost almost all of its cash value. By the computer era the theorem had been verified for exponents into the millions. But millions are not infinity, and mathematics demands infinity.
The winning path came from somewhere else entirely. In 1955 the Japanese mathematicians Yutaka Taniyama and Goro Shimura proposed that every elliptic curve — a kind of equation central to modern number theory — is secretly a “modular form”, an object of exquisite symmetry. It seemed unrelated to Fermat until 1984, when Gerhard Frey observed that any solution to Fermat’s equation would generate an elliptic curve so monstrous it could not be modular. In 1986 Ken Ribet proved Frey’s hunch rigorously. The logic was now clean: prove enough of Taniyama–Shimura, and Fermat falls out as a corollary.
Seven years in the attic
Andrew Wiles had met the problem aged ten, in a library book in Cambridge — his childhood obsession. By 1986 he was a professor at Princeton, and when he heard Ribet’s result he made an extraordinary decision: he would prove it himself, alone, in secret. For seven years he worked in his attic study, telling almost no one but his wife, releasing old papers in slivers to disguise his silence. In June 1993 he surfaced at the Isaac Newton Institute in Cambridge with a three-lecture series under the deliberately bland title “Modular Forms, Elliptic Curves and Galois Representations”. On 23 June 1993, as rumours packed the room, he wrote Fermat’s Last Theorem on the board as a consequence of his work and closed with the driest mic-drop in mathematics: “I think I’ll stop here.” The story made the front page of the New York Times, and People magazine listed him among the year’s most intriguing people.
Then came the agony. During refereeing, Nick Katz found a genuine gap in one crucial argument. For fourteen months Wiles wrestled with it in public view — the worst possible fame for a man who had chosen secrecy. He recruited his former student Richard Taylor, and on 19 September 1994 the fix arrived in a flash of insight: an approach he had abandoned years earlier supplied exactly the missing piece. The two papers appeared in the Annals of Mathematics in May 1995 — 358 years, give or take, after the margin note. Wiles collected the Wolfskehl Prize in 1997. He was 41 when the proof was completed, just past the age limit of 40 for the Fields Medal, so the International Mathematical Union struck him a special silver plaque in 1998, and in 2016 he received the Abel Prize, mathematics’ Nobel-equivalent.
The completed proof runs to well over a hundred pages of twentieth-century machinery. Whatever Fermat glimpsed that day in Toulouse, one thing is certain: his margin was never going to be wide enough.
Quiz nuggets
- Pierre de Fermat wrote his famous margin note around 1637 in a copy of Diophantus’ Arithmetica; his son Samuel published it in 1670.
- The 1908 Wolfskehl Prize offered 100,000 marks for a proof, with a deadline of 2007; Andrew Wiles finally collected it in 1997.
- Wiles announced his proof on 23 June 1993 at the Isaac Newton Institute in Cambridge, ending his lectures with “I think I’ll stop here.”
- Referee Nick Katz found the gap; Wiles and Richard Taylor fixed it on 19 September 1994, and the proof appeared in May 1995.
- Wiles was too old for the Fields Medal (limit 40), so he received a special IMU silver plaque in 1998 and the Abel Prize in 2016.