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.
