Fermat's Last Theorem

“A problem worthy of attack proves its worth by fighting back”

And that's what Fermat's last theorem was doing, it was fighting back every time any mathematician tries to solve it.

So we are talking about Fermat's last theorem I suppose the place to start is with Fermat, Pierre de Fermat, he was a 17th-century mathematician living and working in France. Not working as a mathematician but working as a judge. Every evening he'd go home and math was his hobby, one evening he was looking at an equation which looks a bit like Pythagoras' equation

And he was looking for whole number solutions to that equation, and there are lots of solutions

and so on...

Now Fermat asked himself the question "What if I change this equation so instead of it being x squared, what about if it's x cubed or x to the fourth power?

Are there solutions to that equation? So in general we're talking about

where n is bigger than 2. Does that equation have any whole number solution? He thought about it for a while and couldn't find any whole number solutions, and then he went one step further being cocky he believed he found an argument, he believed he found a proof that showed without any doubt whatsoever there were no whole number solutions. So this is kinda weird, because we have one equation x^2 + y^2 = z^2 that has not just one solution, it actually has an infinite number of solutions. And then we have an infinite number of equations: x^3 + y^3 = z^3,x^4 + y^4 = z^4.....an infinite number of equations which apparently have no solutions.

And Fermat discovered it's proof, and he wrote in the margin of a book he was reading that evening, called the Arithmetica by Diophantus, that “I have a truly marvelous proof but this margin is too narrow to contain.

In other words: "I know how to prove that this equation has no solutions, but I don't have the space to write it down." And then he drops dead.

It very much was a secret proof which he never wrote down

After his death his son rediscovered this book which had this marginal notes "I have a truly marvelous proof which these margins are too narrow to contain". In fact the book is full of these little annoying notes: "I can prove this but I gotta go to feed the cat, I can prove this but I gotta go and wash my hair."

So Fermat was quite annoying in this respect. So his son published a new version of the book the book with all of Fermat's little notes printed in the text. And people would look at these notes and they were like "Fermat said he can prove this, let's try!"

And one by one people rediscovered the missing proofs. And every case where Fermat said "I have a proof", he was right, there was a proof, except in this one example here Fermat's last theorem, it is called Fermat's last theorem because it was the last one that anybody could actually find the proof for. And of course because it's the last one that anyone can prove it's the most precious one, it's the one that's the most desirable.

And the more that people try, the more they fail, the more wonderful it becomes. And this goes on for decades, it goes on for centuries. Right through to the 20th century were people desperate to rediscover what Fermat's proof might have been.

By the time we get to 20th century it's quite clear that this is an incredibly complex problem. Its simple to jot down in a few scribbles what the question is, it's easy to describe the problem.The proof is clearly profound and probably beyond Fermat's reach to be honest.

Some people say Fermat was just fooling around, that it was just a trick, that he left something in his book that he knew would trouble subsequent generations -

But you know there will be always someone who would rise up to the challenge.

And it starts with a ten-year-old child, a child called Andrew Wiles not the one in the above picture though, he's growing up in Cambridge, he went to the library, he got a book called “The last Problem” by E.T. Bell. And the book is all about Fermat's last theorem. And little Andrew Wiles, age 10, decided that he was going to rediscover the missing proof because he was a bright ten-year-old who can understand the problem. But a bright ten-year-old doesn't realize what they're letting themselves in for,

He tried, he talked to the school teachers about the problem, he talked to his A-level teachers about the problem, he goes to university talked to his undergraduate lecturers about the problem. He has a PhD and still this problem is obsessing him.

Yes that’s him, he was about in his late 30's by this time he was a Princeton professor there was something called the Taniyama-Shimura conjecture and trust me you don’t want to get into that. Which had been proposed in the 1950's. So a conjecture is an idea that we don't know whether it's true or not, but somebody is putting it on the table. Somebody proved that there was a link between these two conjectures.

It is much as if you could prove most of the Taniyama-Shimura conjecture you would get Fermat's last theorem for free. So somehow Fermat's last theorem is embedded in this other conjecture.

And Andrew Wiles' childhood passion, his childhood obsession is reignited  because he thinks the Taniyama-Shimura conjecture is worth a go. Wiles didn't tell anybody about it. He worked on in complete secrecy

He started not attending committee meetings, he started going to his office less and less, he started to focus on this problem. Once again not because it was the Taniyama-Shimura conjecture but because it would give him Fermat's last theorem for free. And for 7 years he worked in complete secrecy and at the end of 7 years he suddenly realized that he had Taniyama-Shimura and if he had Taniyama-Shimura, he had a proof of Fermat's last theorem.

He went to Cambridge, he presented his proof on a black board, it was a three-part lecture, the world cheered, he was the front page of the New York Times, he was on CNN he was everywhere. But the sting in the tale is that in any mathematical proof you have to have it checked. You have to have it refereed and published, and when he went through the checking process somebody found a mistake.

Wiles assumed that he could fix it, but the more he tried to unravel this problem the worse it became. And it became a huge embarrassment, you know you've been loathed as the greatest mathematician of the 20th century, you are a hero figure and now you have to admit you made a mistake.

And it took a whole year, but at the end of that year Andrew Wiles working with Richard Taylor managed to fix the proof.

It was like that the Fermat’s last theorem was fighting back. ”A problem worthy of attack proves its worth by fighting back” And that's what Fermat's last theorem was doing, it was fighting back, but Wiles proved that he was too good.

And of course what Wiles proved is that Fermat was right, this equation

For n bigger than 2, has no whole number solutions.