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
x^2 + y^2 = z^2. 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
x^n + y^n = z^n
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:
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 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,
he wrote in the margin in his book, he wrote: I have a truly marvelous proof
which this margin is too narrow to contain.
Hanc marginis exiguitas non caperet 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 have a truly marvelous proof - demonstrationem mirabilem -
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 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 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 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: "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
x^n + y^n = z^n, n 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.