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 exceptions. And this is a little bit
surprising, because, yes, if it's just a little bit bigger
than 1, you get finitely many exceptions. 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 Numberphile. 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.
