It's a fairly natural question. If you're looking at the sequence of prime numbers, so, P1 is 2, P2 is 3, and so on,
how big do these gaps get?
And we know that, actually, the gaps can get pretty big. So there's a very high school argument, that if you take the numbers 1
times 2 times 3, up to, say, times 100, and then you look at this number
plus 2, this is going to be a multiple of 2,
because 2 is a multiple of 2, and
this first big number of product of numbers between 1 and a hundred is multiple of 2. Another way that mathematicians might write this is
100 exclamation mark plus 2.
So this is, multiply all numbers up to 100 together, and then add 2.
But then exactly the same argument means that
100 factorial
plus 3 is going to be a multiple of 3.
And so, 100 factorial plus 2 can't be prime, because it's a multiple of 2, and it's bigger than 2.
100 factorial plus 3 can't be prime,
because it's a multiple of 3 and bigger than 3. And similarly, you can go all the way on to
100 factorial
plus
100, and this can't be a prime, because it's a multiple of 100 and bigger than 100. And so this is
99 consecutive numbers, none of which can be prime numbers. And so you know that
there has to be this gap between prime numbers of size at least 100.
But, obviously, you can replace 100 here with any number you like, and that shows that there are
arbitrarily large gaps between prime numbers.
Brady: "It feels to me you've alrea-, it feels to me you've done a proof, there.
"You've proven that they are..."
Right. So this is completely a rigorous mathematical proof, that whatever number you like, take a million,
take a billion,
there exists a prime gap which is
bigger than a billion, or bigger than a million, or bigger than a billion billion billion, or bigger than the number of atoms in the
universe. And so there are these really huge gaps between prime numbers,
but the prime numbers where these gaps occur may be
really huge themselves.
And then there's this question as to, okay,
if I have a prime number of about a million digits, how big can the
gaps between prime numbers be? For example, if you're trying to find a prime number,
let's say, I challenge you to find, find a prime number that's size about a million.
One way you might try and go about it is,
take a million, try and test if it's prime, if it's not prime,
and a million's not prime, test a million and one, test a million and two, test a million and three, until you find a prime number.
And, this is actually, we believe, a pretty good way of finding prime numbers, that it's completely deterministic,
you can do it quite quickly on a computer.
Maybe takes you a while by hand when they, after the millions, but a computer can do this very quickly.
However,
mathematically, our theoretical understanding of this is
pretty poor at the moment, and it could be the case that you have to go on
ages and ages before you find a prime number, if there were these really large gaps between prime numbers.
Brady: "So you want to make sure that
"random dart you threw at the number line didn't happen to land in the middle of an awesome prime gap."
Exactly. Yeah. So if there was some awesome prime gap, this would completely
screw up your attempt to find a prime number quickly by just testing each number in turn, and it might take heat death of the universe
before your computer pr-, actually finds a prime number, which could be a pretty big problem. So mathematicians are very interested in saying,
given primes of about this size, of size about a million, or about a billion, how big can the prime gaps be?
And, in particular, can we do constructions that are maybe better than this
factorial plus two, factorial plus three type constructions, to try and understand the large gaps between prime numbers.
So I guess this is the large gaps between primes problem, when you're asking
how large can gaps can be.
But, it really became famous because Paul Erdős very much popularized this problem. Paul Erdős is this very charismatic
mathematician. He used to put challenge prizes, so he put numerical bits of money on different problems.
So he would say, if you can solve this problem that I don't know to solve, I'll give you $50. Here's a $20 problem.
Here's a $100 problem.
And he used to do this as a great way of encouraging mathematics amongst the community.
The largest amount of money he put on any problem was
$10,000.
He put it on precisely this large gaps between primes problem.
He didn't want someone to completely solve all we could possibly ask for about large gaps between primes, because that'd be too ambitious.
However, he, and a few other mathematicians, notably, Robert Rankin and
Westzynthius, had a method of doing slightly better than this,
multiply all the numbers up to 100 and then, add two, add three, add five. But it was only slightly better.
And, somehow that's the
only way we knew of constructing what were really larger than expected gaps betwen prime numbers.
It was kind of frustrating for mathematicians that we had this one
way of
constructing large gaps between prime numbers, and we had
no other ways that did any better than random. Erdős' problem was just to try and do better than what they'd done.
So, independently, there were two different proofs of this Erdős challenge problem, using rather different techniques.
But both based on similar ideas to work that had been done previously,
on consecutive days.
It came out on consecutive days.
I worked on this problem.
And also, there was a collaboration between Terry Tao, Ben Green, Kevin Ford and Sergei Konyagin, who came up with a different way of getting,
of solving Erdős' problem, and getting large gaps between prime numbers. So we knew about each other,
but I was the slow one. They got there ahead, one day ahead of me.
We have this website where we put up new papers, and yeah. They got in one day ahead of me.
I was the second one.
So there are different ideas.
So as I said before, to improve on what
Erdős and his collaborators had done, you really needed to get some new arithmetic
understanding about the prime numbers.
In my work, it turned out that you could use some new arithmettic understanding about the prime numbers
that was a generalization of some of the ideas that had been used to work on bounded gaps between primes.
So, somehow, bizarrely, if you reinterpreted the whole ideas behind small gaps between prime numbers,
they could be useful to get a tiny improvement in our knowledge about large gaps between prime numbers, if you took a different
understanding of the whole thing.
Whereas the other paper used ideas based on Ben Green and Terry Tao's previous work on long arithmetic
progressions in the primes.
Quite famously,
around 2005, Ben Green and Terry Tao proved that you get arbitrarily long arithmetic progressions in the primes.
So if you just take one prime, you can
add a constant amount, add a constant amount, add a constant amount, add a constant amount, and you get these strings of prime numbers.
And they showed you can, if you choose those starting prime correctly, and you choose the difference correctly, you can get arbitrarily long
strings of prime numbers. And, again,
reinterpreted in a rather different way, they were able to use this
extra arithmetic information about the primes in the proof of
the ideas of large gaps between prime numbers to solve Erdős' challenge problem,
and do better than what Erdős and his original collaborators had done.
Brady: "Who gets the 10,000 bucks?"
Ah. So.
Um. Uh.
The 10,000 bucks,
is slightly complicated, I think, because Erdős is deceased, but Ron Graham was in charge of it,
and I think he's decided to split it between
the group of us.
Brady: "What's the new state of play?
"What do we now know about the gaps between...?"
So it's a slightly technical statement,
but I'll try and write it out for you.
So, if you look at primes less than x,
so I'm thinking of x as being some huge number, like a billion, or the number of atoms in the universe,
then there is a gap of size, and now we have some big messy expression
involving lots of logarithms.
So log x times log of log of x times
log of log of log of log of x
divided by
log of log of
log of x times
some small constant c
if x is large.
And so this is a slightly complicated and slightly messy mathematical expression for,
if I'm looking at primes of size x, how big a gap can I actively construct?
But this is the current quantity. It's some quantity that gets bigger and bigger and bigger as x gets bigger and bigger and bigger.
And this is the current state of the art.
So, if you want to try and beat our work,
I guess I think you'd need to come up with some new arithmetic information and understanding about the primes.
So the challenge is to do better than this function that's growing very slowly in x.
...but it turns out that that's completely false. That we really don't understand these
guys at all. We really have the most limited theoretical understanding of them. And despite being the most fundamental objects,
most of the most simple questions that you could possibly ask about prime numbers
have been open for thousands and thousands of years, and the most famous mathematicians are...