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卡塔兰猜想 – 译学馆
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卡塔兰猜想

Catalan's Conjecture - Numberphile

今天我要向大家介绍的是卡塔兰猜想
So today I’m gonna tell you about what’s called Catalan’s Conjecture.
虽然它被叫做猜想 但它已经被证明了
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,
这就是一个有关完全幂的问题或命题
and it’s a question about or a statement about perfect powers,
懂了吗?
Okay?
那么什么是完全幂呢?
So, what do I mean by perfect powers first of all?
也就是任取一个整数 如1 2 3等
I mean take some whole number, 1 2 3 so on,
再取一个大于1的整数作为它的指数
and raise it to a power which is larger than 1, okay?
指数可以是2和3之类的
So squares cubes and so on.
如果你把它们列举出来的话
So, if you start to write these things down,
1就是个完全幂
1 is a perfect power,
它的指数可以是任何整数
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.
接下来8也是完全幂
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,
还有16 25等
and 16, 25,
继续这样下去 对 还有27
I gonna run into any cube soon, let’s see, oh, 27,
还有36
and 36,
以及49 这些都是完全幂
and 49 and so on.
所有这些数都由两个整数组成 一个底数 一个指数
So these are numbers that can be built by taking two integers and raising one to the power.
完全正确
That’s exactly right, yeah.
目前为止我们列举了这么多的平方数 立方数
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.
你可以任意取一个大的指数
But you can take the exponent to be as large as you want,
和一个任意大的底数
raise it as high of a power as you want,
也就是说指数和底数的大小是不限的
and you can take the number to be as large as you want to.
然后我把它们列举出来了
I’m just trying to put them in order.
那么我们把这些数字列出来
So do these numbers become more common or less common
可以看出共有的底数是更少呢 还是更多呢?
the further we go down the number line.
随着数字越来越大
So they become less common
渐渐地 你会发现它们更少 对不?
and generally you think they’re spread out, right?
咱们就以平方为例
Like just think about squares, for example.
以一个较大的数字为例 比如说1百万
If you hand me a big number like a million,
你想知道哪些数字的平方少于1百万
and you want to know about how many squares are less than a million,
这时候我们就要算一下1百万的平方根了
well, it’s about square root of a million.
对吗?因为这些数字必须小于1百万
Right? Because that’s how small the number has to be,
被平方的对象肯定小于1百万
to be less than a million after you square it.
这些数字很神奇
They’re cool numbers,
随着数值的增加它们增长的程度也越大
they’re kind of spread out as you go along and get bigger and bigger,
但是要注意数列开头这里
but notice that in the beginning here,
一些数字的数值非常接近
some of them are close together.
从这里我们能得到特殊的一对完全幂
So we’ve got actually a pair of consecutive powers,
这两个连续的完全幂相差1
so separated only by one,
这两个连续的完全幂相差2
I guess here we have separated by two
然后是3和4 以此类推
and three and four and so on,
因此它们之间有一定的差值
so there’s some separation here.
同时 卡塔兰猜想的内容
But what Catalan conjectured
是一个人们千百年来一直很感兴趣的问题
and this is a question people have been interested in for hundreds of years actually,
就是8和9组成的这特别的一对
is this particular pair of eight and nine.
也就是说3²-2³=1
So the fact that three squared minus two cubed is equal to one,
或者说这两个完全幂的值相差为1
or it really that you have any two powers which differ by one,
这种典型例子只此一个
no other example of that was known.
所以卡塔兰猜想的内容就是
And so what Catalan conjectured is that
只有在这种情况下这个等式才成立
this is the only time that this happens
此时这两个完全幂的值相差1
that you have two powers whose difference is exactly one.
但是如果完全幂是天文数字的话就不一定了吧
But it was unknown if like a gajillion and four, a gajillion and five.
对 非常正确
Yeah, that’s exactly right,
所以我们很久以来也不知道
that’s what we didn’t know for a long time,
卡塔兰猜想是否正确
whether or not this conjecture was true.
因此追溯到几十年前
So back a few decades ago we did know that
那个时候我们不知道这个猜想是否正确
if it wasn’t true,
所以只有为数不多的对卡塔兰猜想的证明
there would be only finitely many exceptions.
很容易举出相关例子 对不对?就是这样
So this tends to be easier to show in general, right? That… okay.
因为就算数字是无穷无尽的
I know that there’s a really large number of things happening
但它也是可探索的
or only finitely many times it happens.
我要说的是一个最近的例子
But to show that there’s only one is what we didn’t know until recently.
这个例子很有名 对 是由数学家Mihailescu证实的
So it’s known, yeah, so this is what was proved by this mathematician Mihailescu.
首先 我们要有一系列完全幂
In fact, these are the only consecutive perfect powers, okay?
而这个概念就是:
So this notion that,
随着完全幂的增大 如果差为1的连续完全幂数只有一对
okay, they probably kind of spread out as time goes on,
那么卡塔兰猜想就是正确的
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,
好吧 咱们长话短说
okay, we don’t have time for.
我也不知道卡塔兰猜想在未来几年进展如何
I don’t know, the next couple of years to go through it.
但是我想告诉你们这个例子
But I want to tell you at least about sort of a special case of this,
这个证明曾经风靡一时
and something that’s been known for a while
它给了你一个具体操作思路
but it gives you an idea of kind of the type of manipulation you can do
来证明卡塔兰猜想
to understand this kind of problem.
好的 那么要证明这个方程式仅有一个解
All right, so solving this equation showing that there’s only one solution,
就是这个解 是很有难度的事情
and that’s it, it’s pretty hard.
但是我给你举个例子
But let me show you a special case.
咱们先看这个等式
let’s look at the case of
x²-y³=1
x squared minus y cubed equals one.
比如说 在所有这些连续幂数中
So for example, that has some solution we know,
这个是我们的已知答案
because we have these consecutive powers.
所以我们要求x²-y³=1中未知数的值
So we’re really just asking about a square differing from a cube by one.
好的 那么为什么这个值是唯一的呢?
Okay, so why doesn’t this have any other solutions?
呃 我们这样来做
Well, here’s what you want to do,
一般来说 同时有加法和乘法的定理很难证明
general rule, adding and understanding factors at the same time is hard,
所以费马大定理才那么难证明
that’s why Fermat’s Last Theorem is hard,
而卡塔兰猜想也属于这种很难证明的
that’s why this kind of question is hard.
所以咱们将它转化成一个关于乘法的问题
So let’s change it to a question about multiplication,
好吗?
all right?
所以我们该怎么做呢?
So how do we do that?
咱们来移一下项
Let’s move things to the other side,
所以咱们用移项来解决这个方程式
so let’s say instead, we’re gonna solve this equation.
现在为什么我的心情美好了呢?
Now why have I just made my life better?
原因就是这里分解为两个项了
The reason why is because this thing breaks up into two factors,
那么在这种含平方的等式中
anytime you see a difference of squares like this,
你可以将它因式分解
you can factorize it in this way.
所以这个等式就从一个非常难的关于加法和乘法的问题
And so this just changes from a very hard problem about
变成一个连续相乘的问题
addition and multiplication interacting together to just multiplication.
好的 那么为什么说这样更好呢?
Okay, so now why is this better?
我要怎样找出这个方程式的解呢?
Why do I find out whether or not this has solutions?
答案就在
Here’s the idea,
y这里
the factors of y,
一旦y³除以y 那么这两个因式都得能整除y
any number that divides Y has gotta divide one of these two things.
咱们先暂时不管这两个因式
And so let’s ignore two for the moment,
咱们就来看看y是奇数的情况下
let’s pretend that Y is odd,
这两个因式不能整除y
so two doesn’t divide Y.
那么如果y是奇数的话
So if Y is odd that means then
那就意味着y必须能整除这两个因式中的一个
that any factor of Y has to divide one of these two things,
但是y不能同时除以这两个因式
but it can’t divide both of them at once.
因为无论x等于几 这两个因式的值都相差2
And the reason why is that whatever these two numbers are,
它们仅相差2
they differ only by two.
出于同样的原因 它们没有公因数
So for the same reason, they can’t have a common factor,
应该说不可能有两个公因数
other than possibly two.
所以如果y是奇数 那么要么是y/(x-1)
So if Y is odd, all the factors of Y either divide X minus one,
要么是y/(x+1)
or a factor of Y divides X plus one,
但y不能同时除以两个因式
but not both at the same time,
这意味着这两个因式的乘积必须能转化成某个数字的立方
which means that both of these numbers have to be cubes.
所以这两个因式都得是三次幂
So these are both cubes,
因为不管哪个因式整除y
because any number which divides Y,
此处必须有3个y
we know there’s at least three copies of,
而这两个因式中必须都有y这个公因数
but all of those copies have to go to the same one of these two numbers.
所以这两个因式必须能转化成立方
And so we’ve got two cubes,
它们只相差2 对不对?
which are only two apart, Right?
也就是这两个因式之间的区别是值相差2
So the difference between these two cubes is only two,
但这两个因式都不是三次方
but that never happens with cubes, right?
因为我们知道立方是什么样的
Because we know what the cubes are,
1 8 27等等
1 8 27 and so on.
当然了 你可以继续列举下去
And that for sure spreads out as you go along,
然而你不能找到任何两个相差2的三次方
and so you can’t have any cubes which are separated by two,
所以你找不到这个方程式的答案 至少y是奇数的情况下是这样
and so you can’t have any solution to this equation, at least if Y is odd.
例外呢?
What If it was even?
哦 那就有点难了 但一切皆有可能嘛
Well, then it’s a little bit harder, but not that much.
你还没想到证据吗
You haven’t thought that proof.
不 我有证据 不过通过这个方案找不到证据
No, so in general, of course the proof is not via this plan.
尽管从加法变成乘法的确是
Although changing from addition to multiplication is the important part of the proof actually,
证据链中的重要一环
and important part of the proof.
但这只是
But this is just sort of
通往证实本方程式的
the smallest tiniest piece the step in the right direction
方向正确的极微小的一步
towards proving this kind of thing.
大家好 感谢观看本视频
Hi, everyone, thanks for watching this video.
我负责解决本周Brilliant的相关问题:
I’m just working on brilliants problem of the week,
哪个三角形的面积最大
which triangle has the greatest area.
我不知道这个问题有没有诀窍
I can’t decide if this is a trick question
这些三角形的面积难道不相同吗?
or do they have the same area?
我选择答案C
Let me give C.
啊 它说我回答错了 不过这里有句话说得很对
Ah, it says that I got it wrong but it says here and this is true,
遇上障碍也是学习的一部分
getting stumped is part of learning,
现在我们来探讨一下答案解析吧
and now we get to continue and discuss the solutions.
你知道 我本来可以把这道题做得更好
Yeah, see I probably should have done better working out like that person did,
但我们是匆忙赶制的本视频
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,
你知道么 它还是本视频的赞助者
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
小测验 课程 自创课程
it’s full of all sorts of quizzes and courses and curated lessons
这些课程并非简单地向你普及知识
that doesn’t just tell you stuff so that you’ll know it,
它还会帮你理解 进行深度学习
it’ll help you understand that it helps you go deeper,
我认为这一点才是最重要的
and I think that’s a really important thing.
你可以看大量的视频 并学习大量的东西
You can watch loads of videos and learn lots of new things,
但若要做到真正理解
but to really understand
有时你必须得亲自解决问题
that sometimes you actually have to do the problems
必须有人手把手地教你这个过程
have someone hold your hand as you go through it,
这才是Brilliant的出彩之处
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,
还有科学
they also cover science,
以及电脑科学
they also cover computer science,
物理学 应有尽有
there’s physics, there’s all sorts of things here.
你真的应该去看一下
You really should have a look.
此外 如果你喜欢霍利今天的视频
By the way, if you enjoyed today’s video with Holly,
我会为你推荐Brilliant网站的Open Problems Group版块
I’ll tell the Open Problems Group here on brilliant
这里也是一个能让你不虚此行的好地方
could also be a great place for you to be hanging out.
所以我会在视频下方的简介里
So what are you gonna have a look at that,
附上一张该版块的播单链接
I’ll include a link down in the video description.
现在我们回到这些三角形问题并将之弄清楚
Now while I get back into these triangles and figure things out,
你应该做的就是键入brilliant.org/numberphile
what you should do is go to brilliant. org/numberphile,
这样你就能免费注册Brilliant网站
you can sign up to brilliant for free,
我会把这个链接brilliant.org/numberphile放在简介里
but if you use the slash numberphile I just mentioned
如果你通过链接进入该网站
and there’ll be a link in the description,
那你就能享受8折优惠获得会员资格
you’ll also get 20% off a premium membership.
请去Brilliant网站看看吧 记住不要被动地接受知识
Go and have a look, and don’t just get told stuff.
要理解知识
Understand stuff.
这就是区别所在
There’s a difference.

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视频概述

关于卡塔兰猜想的一些证据以及未解决之处。

听录译者

祐子祐

翻译译者

Licia

审核员

审核员 D

视频来源

https://www.youtube.com/watch?v=Us-__MukH9I

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