Fermat's Last Theorem - Numberphile

Key Takeaways

A problem worthy of attack proves its worth by fighting back. And that's what Fermar's last theorem was doing. It was fighting back. So we're talking about Fermar's last theorem. And I suppose the place to start is with Fair Pier Femar who was a 17th century mathematician living and working in France, not working as as a mathematician, but working as a judge. And every evening he'd go home and maths was his hobby. One evening he he he was looking at an equation um which looks a bit like Pythagoras's equation which I suppose is x^2 + y^2 = z^2. Uh and he was looking for whole number solutions to that equation and there are lots you know there's 3^2 + 4^2. So that's a whole number solution to x2 + y^2= z^2. Now asked himself the question what about if I change this equation? So instead of it being x^2, what about if it's x cubed or x to the 4th power or y 4th power? Are there solutions to that equation? So in general, we're talking about x to the n + y = z to the n where n is bigger than two. Does that equation have any whole number solutions? He thought about it for a while and he couldn't find any whole number solutions. And then he went one step further. Not only could he not find any whole number solutions, he believed he'd found an argument. He believed he'd found a proof that showed without any doubt whatsoever there were no whole number solutions. So this is kind of 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 you have an infinite number of equations. x cub + y cubed= z cubed. x 4th plus y 4th= z 4. An infinite number of equations which apparently have no solutions. And fair discovered this proof. And he wrote in the margin of a book he was reading that evening called the arithmetica by dapantis. And he wrote in the margin of his book he wrote I have a truly marvelous proof which this margin is too narrow to contain. Hank marginous exiguartis non-caparate in Latin. 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. And then after his death, his son um Samuel Clemore, I think, rediscovered this book which had this marginal note. I have a truly marvelous proof. a demonstratium morabulum um which this margins two narrative contain. And in fact, the book is full of these little annoying notes. I can prove this, but I've got to go and feed the cat. I can prove this, but I've got to go and wash my hair. Um so, Fair was quite annoying in this respect. So, his son published a new version of the book, The Arithmetica by Dapantis, but with all of Fair's little notes printed in the text. And people would look at these notes and they would say, "Well, Fair says he can prove this. Let's try it." And one by one people rediscovered the missing proofs. And in every case where fair said I have a proof, he was right. There was a proof except in this one example here. Fair's last theorem is called Fair'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 most desirable. And the more that people try, the more they fail, the more wonderful it becomes. Um, and this goes on for decades. It goes on for centuries right through to the 20th century where people are desperate to rediscover what Fair's proof might have been. Is it widely held that he had done it or had he just claimed like was he telling the truth? I I think by the time we get to the 20th century, it's quite clear that this is an incredibly complex problem. It it's simple to jot down in a few scribbles what the what the question is. It's easy to describe the problem. The proof is clearly uh profound and and probably beyond fair's reach to be honest. Some people say Fairmar was just fooling around. It was just a trick that he he left something in his book that he knew would trouble subsequent generations. I think that's least likely. Some people say that he did have a genuine proof and it's beautiful and it's elegant and it's 17th century and and we could kind of rediscover that proof, but we're just not quite clever enough. I think that's possible, but I unlikely. I think the most likely explanation is that Fairmar thought he had a proof because he was working on his own and because he didn't show this proof to anybody else, nobody could tell him, "Oh, there's a mistake there. You know, line three has got he's got something wrong with it." And and and that's very likely because we know that subsequent generations of mathematicians thought they'd found a proof and then they'd publish it and people would tear it apart and they'd find the error. So what we're looking for is not Fair's proof, which was probably flawed, but we're looking for some kind of proof to see whether Fair was right uh all along. It has a happy ending, and it starts with a 10-year-old child, uh a chap called Andrew Wilds, who was reading a book one day. He he was growing up in Cambridge. He went to the library. He got a book called The Last Problem by ET Bell. And the book is all about Fermar's last theorem. and and little Andrew Wilds, age 10, decided that he was going to rediscover the missing proof. Uh because a bright 10-year-old can understand the problem. A bright 10-year-old doesn't realize what they're letting themselves in for, but but that's another story. And he tries he talks about about his school teachers about the problem. He talks to his A-level teachers about the problem. He goes to university. He talks to his undergraduate lectures about the problem. He does a PhD and still this problem is obsessing him. I think he was about in his late 30s by this time he was a Princeton professor. There was something called the Tanama Shamura conjecture which I kind of think we don't really want to get into at the moment which had been proposed in the 50s. So a conjecture is an idea that we don't know whether it's true or not but somebody's putting it on the table. Somebody proved there was a link between these two conjectures in as much as if you could prove most of the Tanama Shimura conjecture you would get Fermar's last theorem for free. So somehow Fermar's last theorem is embedded in this other conjecture. And Andrew Wilds's childhood passion, his childhood obsession is reignited because he thinks now the Tanama Shamura conjecture is worth a go. You know, he thinks he can get his teeth into that. But but it's still a crazy thing to try and do. And so because it was such an absurd and ambitious challenge, Wilds didn't tell anybody about it. He um worked on it in complete secrecy. He started um not attending committee meetings. He started going to his office less and less. He started to um focus on this problem once again not because it was a Tanya Shimura conjecture but because it would give him Felmar's last theorem for free. And for seven years he worked in complete secrecy. And at the end of seven years he suddenly realized that he had Tanyama Shamura. And if he had Tanya Shamura, he had a proof of Fermar's last theorem. He went to Cambridge. He presented his proof on a black ball. I think 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 it went through the checking process, somebody found a mistake. Wilds 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 lorded as the greatest mathematician of the 20th century. You're 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, uh, Andrew Wilds working with a chap called Richard Taylor managed to fix the proof. I think it's a bit like um the Terminator film I often talk about in terms you know when you just when you think you've slain the monster when you've killed the Terminator he comes back to life and and you have to fight him one last time and and somebody one mathematician I think Pete Hine once wrote a problem worthy of attack proves its worth by fighting back and that's what Fermar's last theorem was doing it was fighting back but Wilds proved that he was too good and of and of course what Wilds proved is that fair was right This equation x to the m + y the n= z to the n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n bigger than two has no whole number solutions. And that's the end of the story. If you'd like to see a bit more from this interview with Simon, I've got some extra footage. I'll put a link in the description. Simon's also got a book about format's last theorem that's excellent. I recommend it. Links below. And just this week, he's got a new book out. Funny about that. It's all about mathematics in the Simpsons. And I think anyone who likes number file is going to love this one. I'll put a link below. But he's also done an interview with me about Ferat's last theorem in the Simpsons, which I think you'll all enjoy. And I'll put that onto number file really soon. But in the meantime, lots of links below. I'll put a link to the WS paper, a few other bits and pieces that I wanted you to see. So uh so have a good look.