Name: 3 as the sum of the 3 cubes - Numberphile
Uploaded: 2020-03-27T18:21:48.000Z
Duration: 13 min 39 s
Description: Thousands of YouTube videos with English-Chinese subtitles! Now you can learn to understand native speakers, expand your vocabulary, and improve your pronunciation...

Past simple

Well, so a few weeks ago, we had a number file video.

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

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.

And before anyone expected it, it just popped out.

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

Representation is the sum of three cubes.

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

This was the problem that kicked it all off.

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.

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

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.

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.

New one for three signed Why you found 40.

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.

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.

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?

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

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

Did that have any of the mathematicians over?

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

This like, you know, this could go forever.

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.

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.

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.

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

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3 as the sum of the 3 cubes - Numberphile

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