ED COPELAND: Hello.
BRADY HARAN: I was going to ask you to describe this in
terms that you would describe it to, say, your daughter.
And then I remembered your daughter does
economics at Cambridge.
ED COPELAND: Yeah.
BRADY HARAN: So let's not do that.
Let's describe this as you should describe to me, maybe.
TONY PADILLA: OK, so there's been a very exciting
breakthrough in the field of number theory.
It caused an awful lot of excitement amongst the
mathematicians, as excited as mathematicians can get.
And the crazy thing about it is that it's come from
somebody who's pretty much unknown.
It's a guy called Yitang Zhang, which is
a pretty cool name.
And he does work at the University of New Hampshire.
ED COPELAND: It's about prime numbers, the things that
certainly got me into maths.
TONY PADILLA: In fact, he really struggled to get an
academic job.
He worked for a time in Subway.
ED COPELAND: There are some amazing properties of primes.
And they've led to lots of conjectures that haven't yet
been proven.
TONY PADILLA: There's nothing wrong with working in Subway.
But normally, these breakthroughs are achieved by
those that are working at Princeton, Harvard, these
kinds of really elite places.
And now we've got somebody who's literally come out of
nowhere, that no one expected to produce this kind of result
and has done something really impressive that many great
minds were unable to do.
ED COPELAND: But one in particular doesn't involve
multiplication of primes.
It involves additions of primes.
And it's the fact that there seem to be an endless series
of primes which differ by 2.
So the obvious ones are the low number primes, so 3 and 5,
and 5 and 7, 11 and 13.
TONY PADILLA: So these two prime numbers are called twin
primes and are called twins because they differ
by this number 2.
ED COPELAND: And then there's a conjecture that goes back
hundreds of years, which says, actually, there's an infinite
number of these.
So the highest known pair is a remarkable, right?
3,756,801,695,685 times 2 to the power of 666,689 plus 1 is
the higher of the pairs of primes.
And if I take away 1, it gives me the lower of
the pairs of primes.
BRADY HARAN: That's epic.
ED COPELAND: It's an epic.
Just to remind you, the lower ones that we were describing
were 3 and 5, and 5 and 7, et cetera.
So to be able to do that and show that that's a pair of
primes that differ by 2 is remarkable.
TONY PADILLA: So these ones that differ by 2 are called
twin primes.
You also get, of course, ones that differ by 4.
These are called cousin primes.
And there's even those that differ by 6.
And these are called sexy primes, which I think you've
done as well.
Why can't you have prime number that differ by 7?
BRADY HARAN: You can't have prime numbers that differ by 7
because one of them will be an even number.
TONY PADILLA: Exactly, Brady.
Well done.
So we know that there definitely are an infinite
number of prime numbers.
And I can prove that for you if you want.
BRADY HARAN: We've done that.
TONY PADILLA: You've done that.
I thought you had.
OK, so you know that there's an infinite
number of prime numbers.
What people aren't sure about is that there are an infinite
number of prime numbers that differ by 2.
But it's believed to be true.
ED COPELAND: And so the goal is to try and show this.
And it's never been shown.
But what has been shown, for the first time, is that you
can bound the difference between two primes.
And somebody has shown-- in fact, Yitang Zhang, from the
University of New Hampshire, has shown that there is a
bound between two primes, let's say one prime a and
another prime b.
And that bound is that it can be some number N. And so if N
would be 2, for the case that we're interested in here-- and
that's the ultimate case that people are interested in.
But what he's managed to show is there is some number N for
which for an infinite number of primes, a and b, this is
going to be less than or equal to 70 million.
BRADY HARAN: So just to be clear, two primes can be
separated by more than 70 million?
ED COPELAND: Oh yes, yes, yes, they can.
But what he's shown is that-- and in fact, the conjecture is
that every single even number, there is an infinite number of
primes that can be separated by that amount.
So here, the even number is 2, right?
So the conjecture is there's an infinite number of pairs of
primes which are separated by 2.
But there's also a conjecture that there's an infinite
number of pairs of primes separated by 4, and an
infinite number separated by 6, and 8, and
in fact, up to infinity.
So that all the even numbers, the conjectures are there are
an infinite number of primes separated by that amount.
But no one has been able to show that's true of any
number up to now.
And what he has demonstrated is there are an infinite
number of primes which will be separated by a number N which
he hasn't yet calculated, but he knows that it's less than
70 million.
TONY PADILLA: There're an infinity of these guys.
[PHONE RINGING]
TONY PADILLA: Oh, god.
BRADY HARAN: What?
Take two.
TONY PADILLA: Hello.
Hi, babe.
I'm in the middle of doing a video.
Well, I've got to answer it so it stops ringing.
All right, call you back when we're done.
All right, see you in a bit.
BRADY HARAN: Was that Ed?
TONY PADILLA: No, it was--
ED COPELAND: The mathematicians who work on
prime numbers will now, no doubt, be scouring over what
he has done and trying to knock this number down.
I mean, I was already hearing about one of the key people
involved, a guy called Goldston, who's talked about
it might be immediately possible to knock this
down to about 16.
And that's a lot closer to 2 than 70 million.
But of course, he has a very nice way of
describing this value.
Maybe 70 million means the primes are not twins, but
they're certainly siblings.
TONY PADILLA: But why is it amazing, I
think, is more the point.
Why is it really incredible?
Well, there's a sort of nice way to illustrate this.
One thing we know is that obviously, there are an
infinite number of prime numbers.
But the gaps between the prime numbers, generically, get
bigger and bigger and bigger.
In fact, you know that for the first N--
for prime numbers between 0 and N, the average gap is of
order log of N. It's a function, but this is a big
number, is the point.
It's not as big as N, but it's a big number.
OK, so let me illustrate what that means in practice.
So imagine you had a scenario where you've got a world with
all the numbers.
And there's some rule--
and I'm just going to impose this rule because I'm king of
this world--
that says that prime numbers can only fall in love with
other prime numbers.
OK, so the idea is that you go on dates with
your nearest neighbors.
And do you fall in love or not?
So for the prime numbers at the lower end of the number
spectrum, they've got it made.
3 gets it down with 5.
7's getting it on with 11.
They don't have to go very far before they
find their true love.
But when you get up to, say, googolplex, in principle, on
average, you expect to go on of order a googol dates before
you're likely to find your true love.
Because the prime numbers are so far apart at that
large end of things.
So it's a pretty loveless place at that end of things.
So you get to bigger and bigger numbers, you might
think there's just no way you're going to
find your true love.
And you probably won't even bother going out of the
house.You'd just stay in and watch
Jeremy Kyle or something.
But what is actually true, though, what Zhang has shown
us, is that for some lucky prime numbers at that very
high end of things, they actually--
and it's always the case-- there are some that actually
will only have to go on about 70 million dates before they
find their true love.
So there are always some prime numbers which are relatively
close together.
BRADY HARAN: 70 million seems such an arbitrary number.
ED COPELAND: Yeah.
BRADY HARAN: And it's like, if it's possible to explain, how
has that fallen out of this proof?
ED COPELAND: 2, 3, 4, 5, 6.
TONY PADILLA: OK, so when people do number theory, how
do they actually go about doing these proofs?
They tend to use sieve theory.