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The Romance of Fermat’s Last Theorem

Fermat left a lot of theorems lying around. Mathematicians proved them all–except one

Pierre de Fermat left behind a truly tantalizing hint of a proof when he died—one that mathematicians struggled to complete for centuries. (François de Poilly, wikimedia commons)
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The story is familiar in the math community and frequently referenced in pop culture: when mathematician Pierre de Fermat died, he left behind him a theoretical math equation and a tantalizing note in the margins of a book. “I have a truly marvelous demonstration of this proposition, which this margin is too narrow to contain,” he wrote.  

It wasn’t the only unsolved theorem that Fermat, born on this day in 1601, left kicking around, but in time it did become the most famous. It was well-known enough that a 10-year-old boy named Andrew Wiles would read about it in a library book in the early 1960s. “I knew from that moment that I would never let it go,” he told PBS many years later. “I had to solve it.”

In pure mathematics, it’s not unusual to devise a theorem with no known proof. In fact, that’s frequently what happens. It’s a little like the fruitless search for the Northwest Passage: explorers knew where the Pacific was, but none of their tries to reach it by an inland passage worked out. However, each try helped map a new part of the continent.

Fermat was  a mathematical genius prone to weird leaps. “After Fermat’s death, mathematicians found a lot of similar notes,” writes Simon Singh for The Telegraph. “I can provide this, but I have to feed the cat” is a memorable one. But over the centuries, all of those theorems were proved, leaving just this one and a three-hundred year history of failed attempts. Writing for The New York Times in 1996, Richard Bernstein explained:

Everybody knew that it is possible to break down a squared number into two squared components, as in 5 squared equals 3 squared plus 4 squared (or, 25 = 9 + 16). What Fermat saw was that it was impossible to do that with any number raised to a greater power than 2. Put differently, the formula xn + yn = zn has no whole number solution when n is greater than 2.

It might look simple, but producing a reliable proof proved to be anything but. “Given that there are infinitely many possible numbers to check it was quite the claim, but Fermat was absolutely sure that no numbers fitted the equation because he had a logical watertight argument,” writes Singh. Whatever it was, we’ll never know, as he never wrote it down.

This is where Wiles comes into the—pardon the pun—equation. Entranced by the three-hundred-year mystery, he first attempted to solve it as a teen. “I reckoned that he wouldn’t have known much more math than I knew as a teenager,” Wiles told PBS.

He didn’t succeed. Then when he was a college student, he realized that he was far from the first to try reproducing Fermat’s watertight argument. “I studied those methods,” he said. “But I still wasn't getting anywhere. Then when I became a researcher, I decided that I should put the problem aside.”

He didn’t forget his first love, but “realized that the only techniques we had to tackle it had been around for 130 years. It didn't seem that these techniques were really getting to the root of the problem.” And at this point, Fermat’s last theorem was nothing new and his interest in it was a bit eccentric.

It took a 1980s mathematical advance to bring the problem into the twentieth century.  Another mathematician proved that there was a link between something known as the Taniyama-Shimura conjecture and Fermat’s Last Theorem. “I was electrified,” Wiles said. He saw that it meant if he could prove the conjecture, he could prove Fermat, while also doing work on a new problem.

He worked on the problem in secret for seven years–then he thought he’d found a reliable proof. When he announced it to the math world  in 1994 it was like saying he’d discovered the Northwest Passage.  (There was an error in his proof, which ultimately he managed to repair with the help of another mathematician.) Today, it’s accepted that Fermat’s Last Theorem has been proven. Last year, Wiles was awarded the Abel Prize (sometimes referred to as math’s Nobel) for his work.

But the question of how Fermat proved–or thought he proved–his theorem remains unanswered, and likely always will. Wiles’s proof is 150 pages long and, he told PBS, “couldn’t have been done in the 19th century, let alone the 17th century. The techniques used in this proof just weren’t around in Fermat’s time. Wiles, like most of the mathematical community, thinks Fermat was wrong. But maybe, just maybe, there is a “truly marvelous” proof out there that’s much shorter than 150 pages. We’ll never know.

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