Name: 最短論文 - Numberphile (The Shortest Ever Papers - Numberphile)
Uploaded: 2021-01-14T10:15:35.000Z
Duration: 9 min 3 s
Description: VoiceTubeの動画で発音を聞きながら英語表現を覚えよう！学べる英語：

[Tony]: I thought we'd have a look at some of the shortest papers

One of the sort of most famous ones was this one here

Which you can literally see, look how short it is!

There's virtually nothing there. But the content's really good

because it actually disproves a very long-standing conjecture

[Brady]: So now we're just disproving with an actual example?

[Tony]: Exactly, they were just proving the Euler conjecture, and what is the Euler conjecture?

Well it's kind of related to Fermat's Last Theorem.

You've got two numbers, a1 to the k plus a2 to the k equals b to the k.

And basically it says there are no integers solutions for k greater than two. Okay, that's Fermat's Last Theorem.

Euler's conjecture was a generalization of this.

Basically, you just take n of these guys, and again, now you're saying,

Euler conjecture would be there are no integer solutions for k greater than n.

No integer solutions for k greater than n, so of course,

Fermat's Last Theorem is the case of n equals two.

But this is not true, right? This paper proves it.

Basically, in this example, n is four, but k is five.

But this doesn't have the record anymore. I don't know if it ever did have the record.

It's sort of a classic short paper that achieves quite a lot.

The record is, I think, currently held by this paper,

by your friend, John Conway, and somebody called Alex Sofer,

I think it is, I think that's how you pronounce it.

Okay, so they ask a question in the title,

And then, so they're basically asking about

how many equilateral triangles you need to cover an equilateral triangle of a given size.

Basically, the paper just says n squared plus two can, and then it has a picture of their solution.

So it's kind of got two words in it, right?

and you imagine it's got a length of side n,

and then you could ask, well, how many triangles of side one,

you know, so unit side triangles, equilateral triangles,

You know the answer. It turns out to be n squared, it's quite easy to see that.

If you've got side two, so we've got side two, we can fill it with four equilateral triangles, like that.

So, these have all got side one and four of them fill the equilateral triangle you started with.

[BRADY]: Which is two squared.  [TONY]: Which is two squared exactly, right?

And it generalizes and the answer is n squared for one of side n.

what if we make it a little bit bigger than n?

Not n plus one but just a little bit bigger than n, so n plus epsilon.

"I know I'm going to need more than n squared."

Because n squared would cover the original one, right?

So I'm gonna need more than, so you're going to need at least n squared plus one.

So they were asking, well can you do it with n squared plus one?

And they didn't actually answer the question.

They say actually you need n squared plus two.

Well, what you do is you sort of stop it at n minus one.

Okay and then you've got a little bit there of one plus epsilon.

So you've got a little of one plus epsilon bit there.

is going to need n minus one squared to fill it. ok ?

right, what I need to fill this little bit left over here?

Well, the way they do it is you know this has got length  n plus epsilon that way.

you're going to need at least n plus one.

Okay, so you squeeze in n plus one of them.

[TONY]: They definitely have to overlap, right?

Because this is n plus epsilon, so it's n plus a tiny bit.

And you're going to squeeze it n plus one of them, so they're definitely gonna overlap,

To fill in the top bits so on and so forth,

In total you've got n minus one squared plus n plus n plus one.

Okay you work out what that is, this is...

n squared minus 2n plus one plus 2n plus one,

[BRADY]: The extra bit you need could be less than two, they're just showing it's--

[TONY]: Yes, so that's just what I think about it. I don't think they even answered the question.

The question is can n squared plus one unit equilateral triangles

cover an equilateral triangle of side n plus epsilon?

They don't say yes or no, they just say n squaredplus 2 can.

Maybe they could have just said "maybe" (laughs).

I don't know, they don't answer the question!

If you look in other fields there are shorter ones there.

It's not maths, but in the applied behavior analysis,

it's "the unsuccessful self-treatment of a case of writer's block".

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最短論文 - Numberphile (The Shortest Ever Papers - Numberphile)

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