[Go on] So here we go I'm going to start here

Well, you might notice that I'm missing out a couple here and there

...29...30...33...34... and so on

Most of them are here but if you look carefully you'll see that here we are missing four and five...

and down here we should be missing thirteen and fourteen; 23 and 22; 31 and 32.

Well, there's a supposition, or a conjecture, that all of these numbers can be written as a sum of three cubes of integers.

Most probably best done by example

If I take this number here for example, 29, it means that I can write this as a sum of three cubes of integers

And in this case it can be written as three cubed plus one cubed, so three cubed is 27, plus one cubed is 28, plus one cubed is 29.

So it's thought that all of the other integers have a similar representation

And I should point out you are allowed to represent your number by negative numbers as well

So you're allowed to take sums of three cubes in which one of the.. or more of the integers is a negative number

What's quite striking is that for some of the numbers, even on this very list,

it's actually surprisingly difficult to write down the solution to the problem

To illustrate what I mean we just have to go a little bit along the line here to number 30, and I'm going to need a little bit more space,

but maybe you would care to hazard a guess as to what the solution to this problem is...?

[Is it an easy one? is it a hard one? ...] -laughing

It's a hard one. I should say that this was only discovered in 1999

It was discovered by computer, it's actually quite surprising I think.

So, are you ready? [I'm ready] OK, let's go

I have had to write this down, but let's go

Two... two, two, zero... four, two, two, nine, three, two... all cubed,

plus minus two, one, two, eight.... [That was a negative number] This is the negative number

And there's one more negative number here: minus two eight three comma, zero five nine comma, nine five six. All cubed

[That's amazing!] Yes it's quite striking I think

[29 was so simple, and 30 was so difficult]

That's right. So, as you carry on up the list you'll find this phenomena continuing

So, some occasions where you'll be able to find very small solutions,

and then just next door to it there'll be a number cropping up which seems to have enormously large solutions

[Are there many other unsolved ones or...?]

Sadly, yes. I'm going to come back to these ones, but the next eligible one is this number 33

And, in fact, we still don't know an answer to that one.

So we've not yet been able to find any integers which when you sum their cubes you can get 33.

Search has been pretty thorough, so far they've gone up to...

I think they've gone up to the numbers of size ten to the fourteen; so it's one with fourteen zeros after it

Within that range, there are no solutions.

[The question people are going to have then is... I mean, maybe this number doesn't belong on the list! Why is the number on the list if we haven't got a solution?]

That's a very good question, and there have been some attempts to prove that this number isn't on the list...

but, those attempts have failed. Therefore it is, and it's perfectly allowable that the first solution might just be ginormous.

[But there are numbers that are off the list]

There are numbers that are off the list. Let me tell you about the numbers off the list

So I'll just write them down again: so we had four, five, thirteen, fourteen, 22, 23, 31, 32.

So you might wonder what those numbers have in common

[What have they done wrong?] -laughing

Well, ah, their great crime is that they can all be written as nine times an integer, so nine times an integer, K, plus four, or nine times an integer K plus five

So for example four is clearly nine times zero, plus four

And for example, 31 here is nine times three, which is 27, plus four

So they all meet this criteria

And indeed, if you write down any number which is of this shape, nine times K plus four, or nine times K plus five

It will never be written as the sum of three cubes. And that's something we can actually prove.

I mean, what's going on here, we're looking at equations of this shape, for example A cubed, plus B cubed, plus C cubed equal to, say for example, 33

So this is an example of what's called a Diophantine equation - and this is a very central topic in number theory

It's been studied, I would say, for lots of years, so they're named after someone called Diophantus, who lived in about 250AD, a Greek

But thinking about problems like this goes back even further, some 4000 years, to the Babylonians

But the question is, if you have a polynomial equation like this, can you first of all decide whether it has solutions in integers - that's what we're trying to do in this special case

If it does have solutions you could even ask how many are there - are there infinitely many, or just, oh you know, just a handful?

And if there are infinitely many, can you describe their frequency in some sense?

And this is the sort of area of research that I'm actively engaged with.

So we do not have a proof that this has solutions, in integers.

It is conjectured that this has solutions in integers

All that we can hope to do, at the moment, is just use a computer and try and find them

[This is not something that can ever be proven by brute force, 'cause there's always another number isn't there?]

Absolutely, yes. Absolutely

So I just wanted to go back to the first number on the list that we had up here, actually, number one.

You know, we're not going to have to spend too much time scratching our heads to come up with a solution to this one.

Right? We could take... you want to have a guess? - laughs [Yep, one cubed, zero cubed, zero cubed]

OK so that was easy. So there's at least one solution, and you might ask, you know, are there others?

I'm going to write another solution: you can write one as ten cubed, plus nine cubed, minus twelve cubed.

So people that have seen your taxi cab video might get a bit excited about that particular one but... let's not go into that here...

[Alright, so you've got... so there are two ways to do this!] Two ways to do this! And in fact there are infinitely many ways to do this in this particular case.

And it's not hard to write down what's called a parametric solution.

So I'm writing one as one plus nine M cubed, all cubed,

plus nine M to the four, all cubed,

plus minus nine M to the four minus three M, all cubed

and the point of this is that this is valid for any M.

Yeah, so in the first case if I put in M equal to zero, I get one, zero, zero

If I put M equal to one, I get ten, nine, and minus twelve.

But you could put anything in, and you'd just get tonnes and tonnes of solutions.

So yeah, in the list that I started with one is a bit special, in that, in fact, it's...

there's this parametric family of solutions, and there are infinitely many different ways of writing one as a sum of three cubes.

There are relatively few examples where we actually have these parametric solutions

And if we go back to one of the examples we saw before, so for example, this number 30

It's suspected that there are not parametric solutions

- although in fact it is suspected that there are in fact infinitely many solutions,

but... they just get big very very quickly

And so as you measure their density there are very few solutions overall.

The whole point about a parametric solution is that there is a formula for the solutions.

You can just plug in numbers and it'll give you some answers

That's not the case for these other examples...

or, it's suspected to not be the case for these other examples - you have to work quite hard to find them.