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.