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. 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 which I suppose is X squared plus Y squared equals Z
squared. And he was looking for whole number
solutions to that equation, and there are lots, you know there's 3 squared plus 4 squared, 5 squared. So that's a
whole number solution to <b>x^2 + y^2 = z^2</b>.
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 <b>x^n + y^n = z^n</b> <b>x^n + y^n = z^n</b> where <b>n</b> 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: not only could he not find any whole
number solutions, he believed he found an argument, he
beleived 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
<b>x^2 + y^2 = z^2</b> that has not just one solution, it
actually has an infinite number of solutions. And then we have an infinite number of
equations: <b>x^3 + y^3 = z^3</b>, <b>x^4 + y^4 = z^4</b>, 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, he wrote in the margin in his book, he
wrote: <i>I have a truly marvelous proof which this margin is too narrow to
contain.</i> <i>Hanc marginis exiguitas non caperet</i>
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 after his death his son, Samuel Clémant I think, rediscovered this book which had
this marginal note: <i>I have a truly marvelous proof -
demonstrationem mirabilem - which these margins are too narrow to
contain.</i> 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 Arithmetica by Diophantus -
the book with all of Fermat's little notes printed in the text. And people
would look at these notes, and they were: "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 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. [Brady]Is that they have widely-held that he
had done it or had he disclaimed, like was he telling the truth? [Singh]I think 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 - I think that's the least likely. Some people say that he did have a
genuine proof and it's beautiful, it's elegant and it's
17th century, and we could kinda rediscover that
proof but we're just not quite clever enough. I think that's possible but unlikely. I think the most likely explanation is Fermat 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: "Ow, there is a mistake
there, that line 3 has got something wrong with it." And
that's very likely because we know that subsequent generations of mathematicians thought they'd found a proof and then
they publish it and people would tear it apart and find the error. So what we're looking for is not Fermat's
proof - which was probably flawed - but we're
looking for some kind of proof to see wheter Fermat was right all along?
It has a happy ending. And it starts with a ten-year-old child, a child called Andrew Wiles, who was reading a
book one day - he's growing up in Cambridge, he wents to
the library, he got a book called <i>The last problem</i> 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 a bright ten-year-old can
understand the problem. A bright ten-year-old doesn't realize what they're letting
themselves in for, but that's another story. And he
tried, he talked about, 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. I think he was about in his late 30's by
this time he was a Princeton professor - there was something called the
Taniyama-Shimura conjecture which I kinda think we don't really
want to get into at the moment - which had been proposed in the 50'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. You know he thinks he can get his teeth
into that. But it's still a crazy thing to try and do and so
because it was such an absurd and ambitious challenge, 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 a chap(?)
called Richard Taylor managed to fix the proof. I think
it's a bit like the Terminator film, I often talk about,
you know when you just think you've slained the monster when you've killed
the Terminator it comes back to life and you have to fight in one last time.
And somebody, one mathematician I think Pete Hines once wrote: <i>"A problem worthy of attack proves its worth by fighting back."</i> 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 <b>x^n + y^n = z^n<b>, <b>n</b> bigger than 2, has no whole number solutions and that's
the end of the story. [Brady]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 Fermat's last theorem, that's excellent, I recommend that, 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 Numberphile is gonna
love this one. I'll put a link below, but he's also done an
interview with me about Fermat's last theorem in the
Simpsons, which I think you'll all enjoy and I'll put
that on to Numberphile really soon. But in the meantime, lots of links below
I'll put a link to the Wiles paper, a few other bits and pieces that I want
you to see, so have a good look.
