Well, I wanted to show you
an interesting sequence of numbers. [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.
Love the coloured Post-it notes. Have a green one, hope it helps with your trauma.
I love his accent. =)
It's Hank Green from another lifetime.
Thanks for the warning!
I must be really tired because I clicked on the thumbnail thinking it was Hank Green in a vlogbrothers video :P
I found the formula for 2, 8 and 16
2 = (1 + 6m3)3 + (-6m2)3 + (-6m3 + 1)3
8 = (2 + 18m3)3 + (18m4)3 + (-18m4 - 6m)3
16 = (2 + 12m3)3 + (-12n2)3 + (-12m3 + 2)3
Prove 2: http://hastebin.com/baqesanavi.parser3
Prove 8: http://hastebin.com/xeyekuneta.parser3
Prove 16: http://hastebin.com/widejimabi.parser3