BRADY HARAN: The last video here
on "Numberphile" was all about gaps between
prime numbers. Specifically, it was about this
paper, which is a proof that's just been published. And what it basically says is
no matter how high you go in the number line, no matter how
high you count, there is always a pair of numbers still
to come with a gap of less than 70 million. What that exact number is,
it doesn't quite say. 70 million sounds like
a really big number. But when you think how big
numbers can get and how rare prime numbers become as you get
higher and higher, it's pretty close. But what's really exciting is
that a lot of people are saying this paper is taking us
a step closer to proving the twin prime conjecture. And what that says is that there
are an infinite number of prime numbers separated
by just two. So that means no matter how high
you count, when you get to a googolplex or Graham's
number or well beyond, there is still an infinite number of
prime numbers ahead of you that are separated
by just two. And when you get that high,
prime numbers become incredibly rare. So to think that all these
pairings are still out there is quite an extraordinary
thought. Now, this proof is 56 pages
long, and it's probably beyond most people. But we did make a video about
it, and I hope you've watched that already. And if you did, there was few
off takes and a few extra bits and pieces from that video that
we didn't use that I've now put into this video. So have a watch of that, and
then I'll tell you what I think this paper at the end. ED COPELAND: I love it. I love it. I get excited about everything,
but I've always loved numbers. And just look at the magnitude
of the kind of things. How on earth does someone
come up with that these are the primes? They're the largest
twin pair primes. And who knows what the
significance of these will be? I love it just for the sake of
mathematics and the numbers. And I'm full of admiration
for people that can this. When I was 18, I had a decision
to make, whether to do physics or whether
to do mathematics. And I went the physics route,
and I'm very happy I did. But if I'd have gone through the
mathematics route, this is the kind of thing I would
have loved to do. But I'm not sure I'm
good enough. Ones that differ by two are
called twin primes. You also get, of course, ones
that differ by four. These are called
cousin primes. And there's even those
that differ by six. And these are called
sexy primes. TONY PADILLA: Now, in principle,
it's believed there's an infinite number
of these twin primes. It's believed there's an
infinite number of these cousin primes. It's believed there's
an infinite number of these sexy primes. In fact, it's believed that
there is an infinite number of each possible even gap
between the primes. But it's not been proven. So what we have here is
a pair of primes. And they differ by 2n. What Zhang has shown is that
there are an infinite number of prime numbers where
this difference does not exceed 70 million. There are an infinity
of these guys. What's even more remarkable,
actually, about the proof that Zhang came up with is that it
didn't actually do anything completely different. It was really just application
of existing ideas but just perseverance. And that's quite unusual. Sometimes when something hasn't
been proven for a very, very long time, it really need
some sort of order and magnitude shift in the
way people think. But this didn't happen here. He just really stuck at it,
perhaps more than others had. It's a pretty impressive
proof. There's no doubt about it. Here it is-- 56 pages of it. So 70 million-- I was talking about potentially
the googolplex, Graham's number, these kind
of crazy numbers. But even then, there's gaps
of only 70 million. So it's a tiny number
in that context. But actually, of course,
it's a lot bigger than 2 or 4 or 6-- it is believed that the methods
that Zhang used could be optimized even more. And you could get this
number down to 16. That's what's anticipated
here. So that would be pretty
impressive. If there's an infinite number
of prime numbers that are separated by only 16 numbers,
that would be really impressive. BRADY HARAN: But not 2. They don't reckon he
can get it to 2. TONY PADILLA: It's not
anticipated that these methods would actually get
it down to 2. 16 is the odds. And it's related to the fact
that a lot of what Zhang did was based on work
of these guys-- Goldston, Pintz, and Yildirim. And they almost got there. They almost got there. What they proved-- well, they proved a
couple of things. One thing that they
proved was that-- so you have this average
gap of log n. So that's the average gap of
between prime numbers. So for example, if n is a
googolplex, this average gap is around the google, is
of that order anyway. So now the question is, what
did Goldston, Pintz and Yildirim show? They showed that you take
any fraction, any fraction you want-- so I take some large n and
you take any fraction you want of this gap-- and I could prove that the gap
between primes is less than. So I take some fraction,
let me call it epsilon. So any epsilon greater
than zero. GPY showed that there exists-- so I'm going to show you a
little bit of mathematical symbol here, that
there exists. That means there exists,
that backward e. There exists prime pairs
separated by less than epsilon log n for larger N. So
you might think, so what is he showing? And this is any epsilon. This just falls short of
proving the conjecture. Now, why does it just
fall short? Well, in principal, I could just
take n bigger and bigger and bigger and bigger. If n's finite, then
OK that's fine. But n can be infinite here.
n can really get big. So that's not enough, because it
just falls short of proving the conjecture, because we're
talking big n here. And n, essentially, you want
to tend n to infinity ultimately to prove that there
are an infinite number of these pairs. You can take this fraction
to be arbitrarily small. But the thing it's multiplying
is getting arbitrarily big. So it's not clear that this
overall size is actually getting smaller and smaller,
can stay small, can even stay finite. It may even become infinite. So they fell just short of
proving the conjecture. BRADY HARAN: Our hero of the
day today has also fallen short of proving the
big conjecture. TONY PADILLA: He least proved
that there was a finite gap, that ultimately there
is a finite gap. And there's an infinite number
of these guys with finite gap. So what GPY did show--
and this is where the 16 comes in-- is they showed that if
something called the Elliott-Halberstam conjecture,
and we don't know it's true. It's expected to be true, but
they don't know it's true. Then there are an infinite
number of prime numbers which differ by less than 16. So this is where the
16 comes in. BRADY HARAN: Just to go back to
our hero of today, what is it about his proof that makes
70 million seem such an arbitrary number, and yet it's
such a perfect round number? TONY PADILLA: So when people do
number theory, how do they actually go about doing
these proofs? They tend to use sieve theory. Now, the simplest version of a
sieve theory comes back from ancient Greece, a guy
called Aristophanes. ED COPELAND: He wanted to work
out what all the primes were, so he began by just writing
out all the numbers. So he begins with 2, because
you know one is not defined as a prime. So he begins with 2, 3, 4. And I'll do 31 and 32. And it goes on and
on, of course. And what he said is
you begin at 2. That's going to be your
first number. And cross out every number
that's divisible by 2. BRADY HARAN: Why? ED COPELAND: Because that
can't be a prime. A prime is defined as a number
that's divisible only by itself, possibly by 1, but
1 doesn't count usually. We begin with 2. So 4 across all the
even numbers now-- immediately wipe out. I'm sieving. They're falling through
the sieve. That's exactly what's
going on. And so I've got rid of that. So now you move to 3, which
is the next one by itself. And you do the same thing. Well, 6 already has
been ruled out. But 9 goes. 12 is already gone. 15 goes, 27 goes, 30 is gone. 4 is already gone. 5 is the next one. So then you start
going through. Anything divisible
by 5 is removed. So 10 is already gone. 15 is already gone. 20 is already gone. 25 hasn't gone yet,
but has now. What you're now left with
actually are all the primes. If I just label the ones that I
left, you've got 11, 13, 17, 19, 23, 29, 31. I've got all the primes
up to 31. And this is the technique that
you can use to actually sieve out the primes. This would give me
all the primes. If I had enough computing power
and enough time and I could do it faster, I would
get all the primes just by writing out this series
and just doing it. BRADY HARAN: And all
infinity of it. ED COPELAND: And all
infinity of it. It took me a long, long time,
an infinite amount of time. But I could do it. BRADY HARAN: Do you know how
long it would take you? ED COPELAND: Infinite
amount of time. So we haven't got that. The funding agencies won't fund
us for that kind of-- BRADY HARAN: We don't have an
infinite amount of time. We've got another 26 minutes. ED COPELAND: What people have
done is they've refined this sieve in such a way that it's
not as accurate as this. So maybe you think of it as a
bit more coarse-grained sieve. And so more things will
fall through. But what you are left with,
rather than having these fine primes immediately left, what
you're left with are groups of numbers with which you
improve your chances of finding the primes. So you have some idea of the
kind of values of numbers you're looking for. You develop your sieve
accordingly. And then all the other stuff
comes through, and you're left with this range of numbers,
within which you hope you're going to find these
big primes. Just look at the sizes that
we're talking about. And so these modified sieves are
used by mathematicians to try and establish these
bounded sets. And the work of Zhang has been
based on some earlier work by a group of mathematicians who
almost got there to find this bound, but couldn't
quite make it. And what he did is he basically
modified the sieve that he was using, which meant
he had just a slightly different way of getting the
bigger prime numbers. And from it, he was able to
demonstrate that indeed there is a bound just based on the
type of sieve he was using. And this bound turned
out to be 70 million that he came up with. BRADY HARAN: So 70 million is an
artifact of the sieve that he arbitrarily chose. ED COPELAND: There
are two things. There's a sieve and then there's
something which I just can't get my head around
called the level of distribution which tells you
how the prime numbers, how regular they become when you're
looking on for very large values. And it was this combination that
he used and out of it, which no one else had
been able to do. And there is a nice aspect
to this story, which is a, he's over 50. And no one had knew about him. He works at the University
of New Hampshire. He'd obviously been working
away, which means you can do great math over 50. You're not totally burnt out. And b, he wasn't that he came up
with some phenomenally new breakthrough. He just kept plugging away at
the techniques people have been doing. And just by doing that he
was able to modify them sufficiently to make this. And he sent it to the
journal in April. And the journal editor,
to their credit-- mathematics journals must be
inundated with people saying they've solved these problems,
because the problems are so easy to say. There's an infinite number of
primes who differ by two. I remember trying to prove this
at school, thinking that can't be that difficult. BRADY HARAN: The "Numberphile"
inbox is inundated with people claiming-- ED COPELAND: I'm sure it is. So to their credit, the
editors of the journal realized this guy knew what
he was talking about. Apparently, the paper
was beautifully written, crystal clear. And they fast-tracked
it through the refereeing process. They must have some way of
putting it on a conveyor belt that sends it to the referees
who all unanimously said this is potentially really
important. TONY PADILLA: I scanned the
proof, but I went online and found some really nice
explanations, because one thing that happened is he
gave a talk at Harvard. And it was extremely
well-received. And people were very
excited about it. Everybody loves prime numbers. So there were people coming
from different fields to see this talk. But what people have done is
some people who were at that talk have put some really nice
explanations online of what the proof involved. So those I find particularly
useful, because they were written by people who weren't
number theorists. And I'm not a number theorist,
so that was very useful to me. Incidentally, there's a nice
little quirk of fate which happened while that talk was
going on, in that a paper came online while that talk was
going on, which actually proved another one of these
crazy conjectures. And it was the Goldbach
conjecture, at least a weakened version of the
Goldbach conjecture. So the original Goldbach
conjecture, which is one of the great problems
of mathematics-- you get one of these
Millennium prizes if you solve it-- is that to prove that every even
number can be written as the sum of two primes. So the weakened version this guy
proved that came out while Zhang was giving his talk was
that every odd number greater than five can be written as
the sum of three primes. So he proved there is a slightly
weakened version of the original conjecture. There's been two really big
events within number theory coming together at
the same time. And the implications for
this in terms of-- now that there's new ideas
coming in, there's even hope that maybe one could solve
the Riemann hypothesis. So the Riemann hypothesis
states that there's a particular function called the
Riemann function, which I can write down quite
straightforwardly. It has the form z through zed is
the sum of n of one over n to the zed. The Riemann hypothesis
asks where are the zeros of this function. It seems what he says
is that they all lie in the complex plane. I'm thinking, where are they
in the complex plane? They all lie on the line with
real number one half. Not obvious-- it's a hypothesis. Great minds have failed to prove
this time and time and time again. And one thing that is hoped is
that now might be progress to proving this. So all these things together,
this kind of hope that's there's going to be
real progress now. Incidentally, we used this
Riemann zeta function in quantum field theory as well. So this has applications beyond
just number theory. BRADY HARAN: You're
a physicist, not a mathematician. Does this excite you when you
read about this and see this? TONY PADILLA: I think everybody
loves prime numbers. I think you can't fail to
love prime numbers. I think the great thing-- I was thinking about this
because I knew you'd ask me that, Brady-- the thing about prime numbers
that I think fascinates anybody who's interested in
math, physics, that kind of thing is that they seemingly
lack some sort of structure. We like to think that we
understand nature, that things ultimately will adhere to some
sort of structure and we could make sense of that. Thus far, prime numbers seem
to sort of evade that possibility. They seem to just be so
higgeldy-piggeldy. What is the structure that
underpins prime numbers? And these sort of conjectures
are sort of going some way towards giving prime numbers
these elusive things that have thus far refused to come on to
the grip of our trying to make sense to them and to apply
structure to them. These sort of conjectures,
improving them is heading some way to doing that. And that's why it's sort of
like, is it an almost final frontier of our understanding
in some way? We're making sense
of something-- everybody can understand why
three is a prime number, a five is a prime number. And it's something that's
familiar to you. And yet seemingly we understand
so little about it. That's the fascination. BRADY HARAN: Do you know what
else a prime number is? TONY PADILLA: What? BRADY HARAN: 19. Stewart Downing. TONY PADILLA: He's has a good
second half of the season, Stewart Downing. BRADY HARAN: Was he
a good signing? TONY PADILLA: I don't want
to say definitely yet. He had a good second
half of the season. I will say that. He's improved. BRADY HARAN: So this paper by
Yitang Zhang, which is, by all accounts, an extraordinary
piece of work. It's well beyond my
capabilities. But I did note it's
56 pages long. And I think the typesetters
have missed an opportunity here. If they just made the point size
a bit bigger or put a bit more spacing in there, they
could have got it to 59 pages. And that's a twin prime. You know that's what
I would have done.
