Science & Technology

# Fermat's Last Theorem Was Scribbled in the Margin of a Book

It seems simple enough: prove that there are no whole-number solutions for the equation xn + yn = zn when n is greater than 2. This is what's known as Fermat's last theorem, and while its origin was quick and informal, it took more than 300 years to solve. Even the mathematician who solved it took his time — he'd been pondering the question since he was 10 years old.

## Math Flex But OK

After French jurist and amateur mathematician Pierre de Fermat died in 1665, his son Clement-Samuel happened upon his copy of "Arithmetic," a math book written by the third-century mathematician Diophantus of Alexandria. Inside, he found a note that Fermat had scribbled in the margin, originally in Latin:

"It is impossible for a cube to be the sum of two cubes, a fourth power to be the sum of two fourth powers, or in general for any number that is a power greater than the second to be the sum of two like powers. I have discovered a truly marvelous demonstration of this proposition that this margin is too narrow to contain."

Given that there's an infinite number of numbers you'd have to check to prove that this was correct, Fermat's demonstration would have to be "truly marvelous." This actually wasn't the only margin-brag Fermat had made — he'd written down plenty of claims like it, and mathematicians soon began working out these so-called proofs for themselves. Most were met with successful proofs, except for one: the puzzle that became known as Fermat's last theorem.

## Try, Try Again

For more than three centuries, mathematicians tried to find a proof for the theorem, and for three centuries, they failed. That is, until Andrew Wiles: a mathematician who had first encountered the problem in 1963 at age 10 in his local library. "It looked so simple, and yet all the greatest mathematicians in history couldn't solve it," he told Simon Singh in the book "Fermat's Enigma." "Here was a problem that I, a ten-year-old, could understand, and I knew from that moment that I would never let it go. I had to solve it."

Three decades later on June 23, 1993, in a lecture hall at the Isaac Newton Institute in Cambridge, Massachusetts, Andrew Wiles took to a blackboard in front of 200 mathematicians and wrote out the proof he'd been working on for seven years in secret, symbol by symbol — and it was wrong. When it was submitted to a mathematical journal, six reviewers pored over every step for two months and eventually found a gap in his logic. The original proof had been front-page news, so the mistake was kept quiet as Wiles scrambled to fix his error. He eventually teamed up with Richard Taylor, his former research student, who worked with him through most of 1994 to close the gap.

Finally, Wiles looked through his original approach one more time and realized the strategy he needed to use — a different approach that he had abandoned years ago. With this new approach, Wiles and Taylor patched up the proof and published their work in May of 1995. This breakthrough was met with lucrative prizes, a knighthood, and a building named after Wiles at Oxford.

Whether Fermat actually had a "truly marvelous" solution or not, it definitely wasn't this one. To prove the theorem, Wiles used advanced mathematics that hadn't been invented until long after Fermat's death. But regardless of Fermat's true aims, the note he wrote in the margin of an old book set the world of mathematics ablaze for 300 years — and let one man solve the problem he'd been puzzling over since he was a child.