• #### 科普

SCIENCE

#### 英语

ENGLISH

#### 科技

TECHNOLOGY

MOVIE

FOOD

#### 励志

INSPIRATIONS

#### 社会

SOCIETY

TRAVEL

#### 动物

ANIMALS

KIDS

#### 卡通

CARTOON

#### 计算机

COMPUTER

#### 心理

PSYCHOLOGY

#### 教育

EDUCATION

#### 手工

HANDCRAFTS

#### 趣闻

MYSTERIES

CAREER

GEEKS

#### 时尚

FASHION

• 精品课
• 公开课
• 欢迎下载我们在各应用市场备受好评的APP

点击下载Android最新版本

点击下载iOS最新版本 扫码下载译学馆APP

#### 卡塔兰猜想

Catalan's Conjecture - Numberphile

So today I’m gonna tell you about what’s called Catalan’s Conjecture.

So even though it’s called the conjecture, it’s actually known.
15年前 一位数学家证明了卡塔兰猜想
So this was proved by a mathematician about 15 years ago,

Okay?

So, what do I mean by perfect powers first of all?

I mean take some whole number, 1 2 3 so on,

and raise it to a power which is larger than 1, okay?

So squares cubes and so on.

So, if you start to write these things down,
1就是个完全幂
1 is a perfect power,

in fact a power of itself to any exponent.
2和3都不是完全幂 而4是的 它是2的平方
2 is not, 3 is not, but then we get 4 which is 2 squared.

Then we have to go up to, I guess, 8,
8是2的立方
which is 2 cubed.
9是3的平方
And 9 is 3 squared,

and 16, 25,

I gonna run into any cube soon, let’s see, oh, 27,

and 36,

and 49 and so on.

So these are numbers that can be built by taking two integers and raising one to the power.

That’s exactly right, yeah.

So of course all we’ve seen so far mostly squares and cubes,
16是个四次幂数
I guess 16 is also a fourth power.

But you can take the exponent to be as large as you want,

raise it as high of a power as you want,

and you can take the number to be as large as you want to.

I’m just trying to put them in order.

So do these numbers become more common or less common

the further we go down the number line.

So they become less common

and generally you think they’re spread out, right?

Like just think about squares, for example.

If you hand me a big number like a million,

and you want to know about how many squares are less than a million,

well, it’s about square root of a million.

Right? Because that’s how small the number has to be,

to be less than a million after you square it.

They’re cool numbers,

they’re kind of spread out as you go along and get bigger and bigger,

but notice that in the beginning here,

some of them are close together.

So we’ve got actually a pair of consecutive powers,

so separated only by one,

I guess here we have separated by two

and three and four and so on,

so there’s some separation here.

But what Catalan conjectured

and this is a question people have been interested in for hundreds of years actually,

is this particular pair of eight and nine.

So the fact that three squared minus two cubed is equal to one,

or it really that you have any two powers which differ by one,

no other example of that was known.

And so what Catalan conjectured is that

this is the only time that this happens

that you have two powers whose difference is exactly one.

But it was unknown if like a gajillion and four, a gajillion and five.

Yeah, that’s exactly right,

that’s what we didn’t know for a long time,

whether or not this conjecture was true.

So back a few decades ago we did know that

if it wasn’t true,

there would be only finitely many exceptions.

So this tends to be easier to show in general, right? That… okay.

I know that there’s a really large number of things happening

or only finitely many times it happens.

But to show that there’s only one is what we didn’t know until recently.

So it’s known, yeah, so this is what was proved by this mathematician Mihailescu.

In fact, these are the only consecutive perfect powers, okay?

So this notion that,

okay, they probably kind of spread out as time goes on,

is it least true if spreading out means bigger than one?
Mihailescu的这个论证思路真的很有潜力
I mean the proof of this is really advanced and so,

okay, we don’t have time for.

I don’t know, the next couple of years to go through it.

But I want to tell you at least about sort of a special case of this,

and something that’s been known for a while

but it gives you an idea of kind of the type of manipulation you can do

to understand this kind of problem.

All right, so solving this equation showing that there’s only one solution,

and that’s it, it’s pretty hard.

But let me show you a special case.

let’s look at the case of
x²－y³＝1
x squared minus y cubed equals one.

So for example, that has some solution we know,

because we have these consecutive powers.

So we’re really just asking about a square differing from a cube by one.

Okay, so why doesn’t this have any other solutions?

Well, here’s what you want to do,

general rule, adding and understanding factors at the same time is hard,

that’s why Fermat’s Last Theorem is hard,

that’s why this kind of question is hard.

So let’s change it to a question about multiplication,

all right?

So how do we do that?

Let’s move things to the other side,

so let’s say instead, we’re gonna solve this equation.

Now why have I just made my life better?

The reason why is because this thing breaks up into two factors,

anytime you see a difference of squares like this,

you can factorize it in this way.

And so this just changes from a very hard problem about

addition and multiplication interacting together to just multiplication.

Okay, so now why is this better?

Why do I find out whether or not this has solutions?

Here’s the idea,
y这里
the factors of y,

any number that divides Y has gotta divide one of these two things.

And so let’s ignore two for the moment,

let’s pretend that Y is odd,

so two doesn’t divide Y.

So if Y is odd that means then

that any factor of Y has to divide one of these two things,

but it can’t divide both of them at once.

And the reason why is that whatever these two numbers are,

they differ only by two.

So for the same reason, they can’t have a common factor,

other than possibly two.

So if Y is odd, all the factors of Y either divide X minus one,

or a factor of Y divides X plus one,

but not both at the same time,

which means that both of these numbers have to be cubes.

So these are both cubes,

because any number which divides Y,

we know there’s at least three copies of,

but all of those copies have to go to the same one of these two numbers.

And so we’ve got two cubes,

which are only two apart, Right?

So the difference between these two cubes is only two,

but that never happens with cubes, right?

Because we know what the cubes are,
1 8 27等等
1 8 27 and so on.

And that for sure spreads out as you go along,

and so you can’t have any cubes which are separated by two,

and so you can’t have any solution to this equation, at least if Y is odd.

What If it was even?

Well, then it’s a little bit harder, but not that much.

You haven’t thought that proof.

No, so in general, of course the proof is not via this plan.

Although changing from addition to multiplication is the important part of the proof actually,

and important part of the proof.

But this is just sort of

the smallest tiniest piece the step in the right direction

towards proving this kind of thing.

Hi, everyone, thanks for watching this video.

I’m just working on brilliants problem of the week,

which triangle has the greatest area.

I can’t decide if this is a trick question

or do they have the same area?

Let me give C.

Ah, it says that I got it wrong but it says here and this is true,

getting stumped is part of learning,

and now we get to continue and discuss the solutions.

Yeah, see I probably should have done better working out like that person did,

but you know we’re recording here in a hurry.
brilliant.org是一个非常出色的解决问题的网站
Now, brilliant.org is a fantastic problem-solving website,

it’s also the sponsor of today’s video in case you hadn’t noticed.
Brilliant网站上充满了各种各样的
And the thing I like about brilliant is

it’s full of all sorts of quizzes and courses and curated lessons

that doesn’t just tell you stuff so that you’ll know it,

and I think that’s a really important thing.

You can watch loads of videos and learn lots of new things,

but to really understand

that sometimes you actually have to do the problems

have someone hold your hand as you go through it,

and that’s what’s great about brilliant.
Brilliant网站上的内容涵盖大量的数学
They obviously cover loads of mathematics,

they also cover science,

they also cover computer science,

there’s physics, there’s all sorts of things here.

You really should have a look.

By the way, if you enjoyed today’s video with Holly,

I’ll tell the Open Problems Group here on brilliant

could also be a great place for you to be hanging out.

So what are you gonna have a look at that,

I’ll include a link down in the video description.

Now while I get back into these triangles and figure things out,

what you should do is go to brilliant. org/numberphile,

but if you use the slash numberphile I just mentioned

and there’ll be a link in the description,

you’ll also get 20% off a premium membership.

Go and have a look, and don’t just get told stuff.

Understand stuff.

There’s a difference.

Licia