Where I reported on the discovery of 42 has a sum of three cubes.

It is?

Oh, yeah.

Yep.

Eso in that video, I said that we're gonna have a go at finding a representation for three.

We're gonna throw everything you have intimacy.

But anything ever once in a generation.

Okay.

And before anyone expected it, it just popped out.

You gotta got it.

Yep.

Three was already known as a sum of three kids wasn't into custody in a couple of ways.

That's right.

Should I write that down?

Three had to known.

Representation is the sum of three cubes.

One cubed plus one cubed plus one cubes.

And also, uh, four cubed plus for cubed plus negative five cubes.

So both of them single digits.

This was the problem that kicked it all off.

So it was in 1953.

That more Del asked kind of as an offhand remark, actually, in the middle of a paper about something else, he says, I don't know anything about the solutions of this problem, you know, other than that too easy ones, and it must be very hard indeed to say anything about other possible solutions.

That turned out to be an understatement, because it took 66 years to find the next one, and it has a couple numbers with 21 digits.

Two of them have 21 digits and the other one has 18.

We should point out there are some numbers that can never be found.

That's right.

So if your number, if you divide it by nine and you gonna remainder of 45 then we know you can do it.

So 45 13 14 etcetera.

But the others it's expected.

There's a conjecture that says that there ought to be infinitely many representations of the sum of three cubes, but now it's not proven well, so there are a few numbers for which it is Andi.

Think Tim talked about this in the very first video, you can write down Parametric families for the number one or two.

Okay, but starting from three on Benny numbers have no such paramedic families that we know of and we don't expect them.

Nevertheless, we think that there are infinitely many solutions that just kind of sporadic, so the conjecture is every number other than that special class.

He has infinite number of ways to express it.

A sum of three.

That's right, and there's even kind of a conjectured density.

So we expect, you know, there's a formula for how many we expect, roughly up to a certain number of digits.

What's special about three is that it has the lowest expected density of any number of 200.

So it's not so surprising that this one was hard.

Nevertheless, it we did have to go much further than we would have expected to find the third solution.

And just quickly, How did you find this?

New one for three signed Why you found 40.

Yeah, pretty similar.

There are it Turns out there's some optimization that are specific to the number three that we could do and more or less that let us go Another order of magnitudes for free over what we could do for 42.

And so it was about the same size computation.

A cz for 42 you use this big network.

Is charity Internet?

Yeah, we use charity engine ends.

To be honest, we have yet to throw anything at the charity and you guys that caused them to break a sweat s o.

I sent an email with a new version of the code around noon on a Monday and seven hours later, I got this in the email with the solution.

Partly that was down to luck.

But also partly because, you know, charity and his network is big.

I was in band rehearsal and, you know, just during break time, I checked on my phone on So what?

Their hands, uh, was really excited.

I'm pretty sure my bandmates think I'm a complete nutter.

Now, did you tell?

What did you tell you about my Well, not all of them.

Just the ones near me.

I tried to explain a bit.

Yeah.

Did that have any of the mathematicians over?

They lie?

No.

Said they were just like Okay.

Yeah.

Whatever, Andy.

Yeah?

Yeah.

Why is this significant?

It seems like you mean you just knocking off one number at a time.

And one new solution of time.

This like, you know, this could go forever.

Why was three worth even doing?

That's right.

Well, I mean partly it's for historical reasons, because that this is the problem that kicked off the whole thing, and it's very satisfying to be able to solve this.

There are theoretical reasons, too.

So this touches on something called Hilbert's 10th Problem.

So Hilbert ce made a famous list of 23 problems at the beginning of the 20th century.

You know, kind of outlining his vision for mathematics in the coming century.

Number 10 on the list.

Waas to find a way of telling if any given dia phantom equation has a solution or not.

So that's just a polynomial equation on you're looking for injured your solutions.

At first glance, this seems pretty reasonable.

So in school you wouldn't have the soul of linear equations and quadratic equations University.

You take it up to any number of variables, but it turns out so.

This is one of Hilbert's problems that solved now, but probably not in the way that Hilbert envisioned it all.

So it's solved in the sense that we know it's impossible.

Okay, so for General Dia Frampton equations, uh, it's not always possible to solve them.

So there could be Dia Frampton equations out there that have no indeed your solutions, but also no way of proving that fact, and what makes this example of the sum of three cubes really interesting is it's right at the sweet spot for being a difficult problem along these lines.

On the one hand, it's very close to things that that are relatively easy, that we understand.

For instance, if instead of three cubes you had three squares, right then we have a complete characterization of those numbers that could be written as a sum of three squares effect.

That's a famous theorem from the end of the 18th century of genre, or if, instead of three cubes, I'd two cubes.

And then again, we have a characterization of those numbers on Like Wise.

I think if you have more cubes than it's not completely solved for four cubes, but there's there's lots of progress.

It cz three cubes where it really turns out to be difficult.

And really nobody has any idea how to attack this theoretically, and I can see why 42 was fun to do it, because it's 42 it was the last one below 100.

Yeah, and I can see why three was worth doing because it has this historical nice and story started.

But for how long?

Can you just keep picking off these solutions that mean Wendy Windows, All the attention turn toe a concrete proof.

That's right.

So, yeah, the problem is for Given all that we know now, this might be the only thing that we could do.

So as far as anyone knows, that could be a doll Fenton equation of this type.

For instance, 114.

That's the next number we don't know.

Does it have a representation of the sum of three cubes?

It's conceivable that the answer is no, but also that there's no way to prove that.

So it's a little dissatisfying, but you know, that could be the case, and we don't think that's the case.

We think that if you look long enough, um, that you will find a solution, but nobody knows for sure.

And even that in itself would be a non decided well, problem does every number that we think should have representation has it a representation.

I always get very confused when I go, you know, down this rabbit hole, it goes very deep.

But where do you start?

You searching for 114 way are going to.

So you know, every time I think we're finished with this, then we find it.

We think of a few more things that that weaken.

D'oh!

So we are gonna have a go.

Actually, I haven't said yet.

We found another find representation for 906.

So that knocked another number off the list down to nine.

Up to 1000.

But I should say, I'm I'm quite skeptical that we're going to solve all of the nine or many ones up to 1000 anytime soon.

But is there a point just continually knocking these off the list?

If you think they've all got one anyway, shouldn't all the energy be going into the proof?

Are you working on the proof that not your area?

That's not my area.

I am somewhat sympathetic to this.

Yes.

So the point is to go until you're pretty sure of what is true.

I think we're at that point for a while.

That wasn't always the case, right?

There was maybe 30 years where many people thought that maybe this wasn't true.

You know that there weren't any more representations for three or there were Denny for some of the other small numbers.

33 42 center.

Now we're pretty sure that that hate Brown's conjecture That says there are infinite.

Minnie is true.

We have good evidence for that now.

And so at least we're trying to prove the right thing.

You probably saw in the last video.

Lots of people say this is cool, but what's what's the point of to keep doing this?

Why are you keeping doing it?

You say you're pretty confidence, Truth?

This computational power could be used for other things.

Why keep knocking the numbers off the list just for completeness s.

So we are gonna stop eventually.

Mostly.

It's just a bit of fun.

Yeah.

And the historical reasons for three.

And the numbers up to 100.

That's right.

And you want to finish that 1000?

Honestly, Brady, I'm not sure that I'll live long enough to see that.

That that that wasn't done.

Really?

Yeah, well, I've had discussions about this with my daughter, too.

Yeah, it's a bit fatalistic, but yeah, she might live to see it, but but I probably won't I don't know.

Last time we spoke about three and within days you'd found why did the remaining nine ones below 1000 have to be so unobtainable?

Couldn't they just be a similar number of digits, or do you think one of them has to have?

It's possible, But so I talked about this conjecture density, right?

And for some of these numbers, um, I have to look up.

I think it was 390 that has the lowest density.

It's really small, Um, and if you haven't found one by a certain point, well, it's kind of like, you know, if you expect your bus every 10 minutes, if it's supposed to come every 10 minutes, said, you've been standing there 20 minutes at the bus stop, you still expect to wait another 10 minutes, right?

It's that kind of thing.

So, um, where it's very likely that we'll have to go another 78 you know, maybe more digits further than we have up to up to now to find a representation for 390.

Where's the next three likely today next, one for three that also I'm not expecting to see anytime soon.

Yeah, so we expect the solutions for three on average to get about seven digits longer with each new solution, I notice.

Always they three numbers always have a similar number of digits.

Is that so?

This one was a little peculiar.

So, uh, it was quite extreme.

Well, it was very skewed.

So the 1st 2 numbers, we're nearly equal in size, but opposite in sign so that they almost cancel each other out.

And then the third variable with small right, just to make up the difference.

That happens a fair amount of time, actually.

So one feature of the new algorithm that we're using, as opposed to earlier attempts, is that it only matters what the size of the smallest number is, right?

So if we had to do a search that went all the way out to finding the biggest numbers, then we wouldn't have been able to do it.

The technology we have now is it possible there's a solution for three with smaller numbers that you've brushed past.

It's possible depends what you mean by smaller.

So there shouldn't be any with the smallest number smaller than the one we found.

But because the 1st 2 numbers are so enormous in the solution that we found, it is conceivable that there is a solution where the total digit size over old the numbers is smaller.

Yeah.

How are you feeling about this?

I feel pretty good.

Yeah, it's quite satisfying, Andi.

I think we're nearing a sense of closure on this problem.

But let me turn it around as how you feel.

Ready?

Well, I know the number before video like nudged you to look into this, So I feel like like like some sort of no point, no involved.

So you come on, you're being modest.

So I would say that number files contribution has been as important a cz mine or anyone else's who's worked on this.

So, in fact, each video gave us just the boost that we needed to tackle the next challenge, starting with the 1st 1 you know, with Browning.

That's a very good question that happened.

Some tends to prove that this number isn't over this, but that got some Sander who spends attention who read a bigger search and found 74 Solution one.

And then it was that video on that 1 74 that really got me into it and found 33 we found three years cube, some 33.