A note in a book margin took mathematicians 358 years to prove.
Around 1637 Pierre de Fermat scribbled in the margin of a book that he had a marvellous proof, too large to fit, that one simple-looking equation has no whole-number solutions. Nobody ever found his proof. Andrew Wiles finally cracked it in the 1990s, after six years of secret work and one near-disaster.
Everyone knows some whole numbers fit a² + b² = c², such as 3, 4 and 5. Fermat claimed that if you raise the powers to 3 or anything higher, there are no positive whole-number solutions at all. He wrote this in his copy of the ancient Greek text Arithmetica, adding that the margin was too narrow for his proof. His son published the notes after Fermat's death in 1665. It became known as his 'Last Theorem' because it was the last of his claims left unproven.
For centuries it resisted everyone. Leonhard Euler handled the case of cubes; Sophie Germain devised an approach that covered a whole class of primes; Ernst Kummer proved it for a large family of exponents in the 19th century. In 1908 the industrialist Paul Wolfskehl left 100,000 gold marks as a prize, and thousands of incorrect proofs poured in. The problem earned a reputation for attracting more wrong proofs than any other.
The breakthrough came from an unexpected direction. In the 1980s Gerhard Frey and Ken Ribet showed that Fermat's theorem would follow from a deep conjecture linking two separate areas of maths, elliptic curves and modular forms. Most experts thought that conjecture was out of reach. Andrew Wiles, fascinated by Fermat since childhood, worked on it in near-total secrecy for six years and announced a proof in June 1993 in a series of lectures at the Isaac Newton Institute.
Then a flaw emerged in one step. Wiles spent about a year trying to fix it, joined by his former student Richard Taylor. On 19 September 1994, close to giving up, he saw how to combine two failed approaches into one that worked. The corrected proof was published in 1995, 358 years after Fermat's note. Wiles later received the 2016 Abel Prize. His proof rests on 20th-century mathematics far beyond anything Fermat knew, which fuels long-standing doubts that Fermat ever had a correct proof of his own.
Source: Wikipedia — Fermat's Last Theorem · Text summarised from Wikipedia (CC BY-SA 4.0)
