abc Conjecture - Numberphile

Uploaded by numberphile on 12.10.2012


A proof has been announced of a unsolved, so far, conjecture
called the abc Conjecture.
If this proof is right, then it's going to be news on the
scale of Fermat's last theorem was in the '90s, which was
this big unsolved problem.
And was this huge event.
So it's really exciting.
We don't know if this proof is right yet, so a Japanese
mathematician called Mochizuki has released these papers, and
altogether, it's 500 pages long.
He's been working on this for a very long time.
He's come up with his own theory of maths-- a whole new
body of maths--
and he's called it interuniversal geometry.
I know nothing about that.
Very few people do.
Even the experts don't know much about this at the moment.
So it's going to take a very long time to make sure if the
proof is right, because they're going to have to learn
this whole new theory of mathematics.
So the abc Conjecture involves the most simple formula you
can think of.
It's this--
a plus b equals c.
It doesn't get much easier than that.
And that's where it gets its name from.
The rules are, these are whole numbers, and they don't share
any factors.
So that means if I can divide a by 2, then I'm not allowed
to divide b by 2.
Or if I could divide a by 3, then b is not allowed to be
divisible by three.
They're not allowed to share any factors like that.
All right, let's try an example that works.
1,024 plus 81 equals 1,105.
Right, now let's just check they don't share any factors.
In fact, I've picked these on purpose.
This one is 2 to the power of 10, and this one is 3 to the
power of 4.
So they don't share any factors there.
Oh, and this one, I'll do the same, is 5 times 13 times 17.
Now, this is what I want you to notice.
On the left hand side, you've got lots of prime numbers.
We've got all 10 of them over here, and another four.
Loads of them over here.
On the right hand side, you only have three, and this is
what you tend to see most of the time.
This is what's normal.
If you get lots of primes on the left, you only get a few
on the right hand side.
So this is what the conjecture is about.
I want to show you one where it doesn't work.
3 plus 125.
That's equal to 128.
Let's just check they don't share any factors then-- well,
that's 3, this is 5 cubed, and this, 128, is 2
the power of 7.
Now, this one is not like the first one I showed you.
You've only got a few primes on the left, but we've got
loads more on a right.
So you've got more on the right than you do on the left.
That's unusual.
That's weird.
So rather than not working, like I said earlier, it's the
unusual example.
These don't happen so often.
So the technical way to say this is this.
Times those primes together.
So I'm going to do that.
So 2 times 3 times 5 times 13 times 17, and
that equals to 13,260.
It's a big number, and it's bigger than the right hand
side, which was this.
That's what normally happens.
OK, so if you do this, you get a bigger number.
I'm going to show you this one that I said was unusual.
If we do the same thing--
3 times 5 times 2, that's equal to 30, and that's
smaller than 128.
So that's the difference.
So this is unusual.
This is much smaller than the right hand side.
This number, when you multiply the primes together, is called
the radical of ABC.
It's called the radical because it is [INAUDIBLE].
The abc Conjecture is the radical--
which I told you how to work out, that's this--
the radical of abc is bigger than the right hand side.
I said that was c.
That's what you get normally.
In fact, the conjecture is more than that.
It talks about the powers of that, too.
But there are exceptions, and these are the exceptions.
When k equals 1-- that's the power is 1--
there are infinitely many exceptions just like the one
I've just shown you there.
Infinitely many, even though I said these were the rare ones,
the unusual ones, there are infinitely many of them.
But if you take a power bigger than 1, even if it's only a
little bit bigger, even if it's like a power of 1.00001--
tiny, tiny little bit bigger--
if it's bigger than 1, then you get finitely many
And this is a little bit surprising, because, yes, if
it's just a little bit bigger than 1, you get finitely many
You could count them off.
You could write them down.
You could say, here are all the exceptions for this power.
And that's unusual.
That's unexpected.
Now, this is the conjecture.
It's very abstract.
It's very pure.
This was made in the '80s, this conjecture.
But if this can be proven, what it's going to do is it's
going to prove a whole bunch of other stuff at a stroke.
And that's why it's big news.
Originally, they thought that Fermat's last theorem, which I
talked about being solved in the '90s, they thought this
was the way to solve it.
Because there is a way that, if you can solve this, you can
solve a version of Fermat's last theorem.
It didn't turn out that way, because Fermat's last theorem
was solved first.
I heard of it, I think, before it went around the nerdy
blogs, and I thought, well, we could talk about it on
But then, it's still not been checked, so maybe we shouldn't
talk about it on Numberphile.
But then when all the blocks started going mad about it, I
thought someone might ask us.

I mean, that's how you probe extra dimensions.
That's how you probe the very small.