Mathematician Andrew Wiles of the University of Oxford was awarded the prestigious Abel Prize for his remarkable proof of Fermat’s Last Theorem in the early 90s.

Wiles won 6 million Norwegian Kroner as part of the prize, equivalent to about $700,000.

Fermat’s Last Theorem was originally suggested over 350 years ago, but Wiles’ proof of the theorem involved proving a more general result in modern algebraic geometry, bringing in complex mathematical techniques that were not developed until the 20th century.

Despite the complexity of the proof, the theorem itself is pretty straightforward. Here’s the fascinating problem that Wiles solved that led to his prize.

## Perfect squares and cubes

Perfect squares are whole numbers that are the square of some other whole number. 25 is 5^{2}, 64 is 8^{2}, and 81 is 9^{2}.

It turns out that some perfect squares are in turn the sum of two other perfect squares: 25 = 9 + 16, or 3^{2} + 4^{2}. Indeed, there are infinitely many such perfect squares, and a perfect square along with the two other squares that add up to it are called Pythagorean triples for their geometric relationship to the Pythagorean Theorem regarding right triangles.

Of course, not all perfect squares can be written as the sum of two other perfect squares — a few minutes of playing around with, say, all the ways to add together two whole numbers to get 9 will show that there’s no pair of perfect squares in that list.

When mathematicians see a phenomenon like Pythagorean triples they often jump pretty quickly to considering generalizations of that phenomenon. In our case, we want to know if we can extend the idea of Pythagorean triples to higher powers: Could I take a perfect cube, like 8 (which equals 2^{3} or 2 x 2 x 2), and write it as a sum of two other perfect cubes? What about a perfect fourth power?

Formally speaking, for a power n bigger than 2, can I find whole numbers a, b, and c so that c^{n} = a^{n} + b^{n}?

Wiles won the Abel prize for his 1994 proof that the answer to that question is an emphatic “no”: Triples like this don’t exist for powers bigger than 2.

Given that this is, on the surface, a fairly simple question, it’s remarkable that the answer wasn’t proven for over 350 years since it was first posited.

## Fermat’s Last Theorem

This mathematical statement — that we can’t take perfect whole number powers and break them into sums of other perfect powers — is known as Fermat’s Last Theorem after the 17th century French lawyer and mathematician Pierre de Fermat.

The theorem is named after Fermat because of an amazing note written in the margin of his copy of a classic Greek maths text around 1637, as per Wikipedia (emphasis ours):

“It is impossible to separate a cube into two cubes, or a fourth power into two fourth powers, or in general, any power higher than the second, into two like powers.

I have discovered a truly marvellous proof of this, which this margin is too narrow to contain.“

Of course, Fermat never actually published this supposed “truly marvellous proof.” Given that Wiles’ proof of the theorem used elaborate mathematical techniques that didn’t exist until the second half of the 20th century, many contemporary mathematicians are sceptical that Fermat had an actual proof.

This is not to say that the estimable lawyer was trying to commit fraud. It’s far from uncommon for a mathematician to convince him or herself that they have cracked a problem only to later realise (or for their peers to realise and inform them) that they missed some key, subtle detail or another that derails the whole project.

Regardless of whether or not Fermat actually had a proof, his method of claiming a solution that he didn’t write down because he didn’t have enough space has definitely tempted many maths students confronting homework deadlines in the centuries since.

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