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7825问题

The Problem with 7825 - Numberphile

近来公布了一项新证明
A new proof has been announced
号称有史以来最庞大的证明
which they claim is the largest proof ever.
它分为两部分
It comes in two parts.
第一部分在这里
And the first part is this.
这就是第一部分
This is part 1.
第二部分足有200太字节大
And part 2 is 200 terabytes large.
这可太大了!
That is huge!
我会从简单些的部分开始
I’m gonna start with something simpler,
也许你也可以跟我一起来试试
something perhaps you can try yourselves.
好 我的想法是这样的 先取数字1到9
Okay, this is the idea.Let’s take the numbers 1 to 9.
我准备把它们填进九宫格里
I’m just going to draw them out in a grid, here.
假设这些格子是1到9我先不填数字进去
Imagine these are the numbers 1 to 9. I’m not gonna write the numbers in quite yet.
我现在感兴趣的是 a+b=c
And now I’m interested ina+b=c.
这些都是整数 a+b=c我也写在这儿了
So these are gonna be whole numbers, and I’m gonna have a+b=c.
这个式子并不难算
That’s not a difficult equation.
再考虑点儿不一样的 a<b<c
I want it to be different, so let’s have a 如果只用红色和蓝色给1到9着色
Can I color these numbers, 1 to 9, using red and blue,
能否保证a+b=c中的三个数字不全同色?
so that I don’t have a+b=c all the same color?
我不想它们全为红或全为蓝 我得避免这个
I don’t want them to be all red or all blue.That’s something I want to avoid.
那么我们来试试能不能做到吧
So let’s try to see if we can.
先从用红笔写1开始吧 这没什么问题
Let’s start with 1 as a red.There is no reason why not.
然后我们来用蓝笔写个2
And let’s have 2 be a blue.
我们知道 3等于1+2
This means, 3 is 1+2.
没事 我们可以随便选一个颜色 红蓝都可以
That’s okay. We can pick a color. It could be red or blue.
Brady 你来选吧 你觉得红色怎么样?
And Brady, you can pick if you want. Do you want to be red?
-我们选红色呗 -好啊 没问题
– Let’s make it red.- Yeah, okay, fine.
那么我们来写个红3
So we’ll have 3 in red.
嗯 有趣 现在我们就可以做一些推算啦
Well, that’s interesting because now we can make some deductions.
3是红的 1是红的 所以4不能是红的
3 is red, 1 is red. So 4 can’t be red.
这是我们要避免的情况
That’s what we are trying to avoid.
我们不希望它们仨都是同一个颜色
We don’t want them all to be the same color,
因为那样的话a+b=c就同色了
because then we would have a+b=c all the same color,
或者说1+3=4会全是红色的
or 1+3=4 would all be red.
这就是我们要努力避免的
That would be something we’re trying to avoid.
也就是说 4必须是蓝色的
That means, 4, it’s gonna to be blue.
用蓝笔写4 好!
Let’s write 4 as blue, great!
还有些东西我们也可以用来继续推算
Because that’s something we can use as well.
2是蓝的 4是蓝的 6就必须是红的
2 is blue, 4 is blue, 6 will have to be red.
我们用红笔把6写好
Let’s put 6 as red.
现在有意思了 因为已知1+6=7
That’s interesting now. Because we’ve got 1+6=7.
我们不能给7着红色 它必须是蓝的
We’re trying to avoid it being red.It’ll have to be blue.
那就用蓝笔来写
Let’s make it blue.
6+3都是红的 所以我得用蓝笔写9
6+3, those are both red, so I’m gonna have 9 as blue.
好 目前我们做的挺好 还没什么问题……
Okay, we’re doing okay so far, don’t have a problem…
啊 等等 这有个问题
Oh, but wait. There is a problem.
2+7=9
Because 2+7=9.
但是这三个数全是一样的颜色
And they’re all the same color.
噢 不 我给搞砸了
Oh, no, I messed up.
我没做好 在这儿出问题了 已经彻底失败了
I didn’t do it, I failed here, this has failed.
数字1到9是有很多其他的着色方式的
Well, there’s lots of other ways of coloring in the numbers 1 to 9,
实际上 它们各有两个选择 红色或蓝色
In fact, each of them has two options, red or blue,
所以我们这样算 2的9次方等于512
so it’s going to be 2 to the power 9, 512.
一共就是512种着色方法
So there’s 512 ways you could color these in.
你总是会失败的 这一点并不难理解
It’s not very hard for you to convince yourself that you’re always going to fail.
每次的结果都会是失败
You’re always going to end up with something like that.
你也可以验算一下所有的方法
Or you could just check, all the options.
如果你乐意 当然可以验算这512种方法
If you want, you could check the 512 options.
然后你会发现避免a+b=c是同种颜色
And you’ll see that it’s not possible
是不可能做到的
to avoid a+b=c in the same color.
该问题讨论的就是这个 但他们研究的更深入
That’s what this problem was about, but they took it one step further.
他们关注的是a²+b²=c²这个式子
They were interested in a²+b²=c².
他们也和我们一样
And they wanted to color in the numbers using red and blue
想用红和蓝给数字着色
just like we did before.
但他们要避免的是a²+b²=c²同色
But they wanted to avoid a²+b²=c² being the same color,
即全红或全蓝
so all being red or all being blue.
你可能知道a²+b²=c²是什么
Now you might recognize a²+b²=c².
这就是毕达哥拉斯定理(即勾股定理)
That’s Pythagoras’s theorem.
但是我们只关注这个问题的整数解
But we’re interested in whole number solutions for this.
其中一个解可以是3²+4²
So a solution might be 3²+4².
其实就等于5² 不如我们来验算一下吧?
That’s actually equal to 5². Shall we just check and see if that’s a solution?
3²=9 4²=16 加起来就是25
3²=9, 4²=16. Add them together is 25.
好嘞 25=5²
Great! Becase 25=5².
5²+12²=13²也可以
Another solution to this might be 5²+12²=13².
这是另一个整数解
That’s another solution that just uses whole numbers.
这些都叫作毕氏三元数
These are called Pythagorean triples.
他们并不少见 其实古希腊人就知道如何构造毕氏三元数了
They’re not particularly rare.In fact, the ancient Greeks knew how to make them.
你想自己试试的话 我会教你怎么构造的
I’ll show you how to make a Pythagorean triple, if you want.
取两个数字 M和N
You just take two numbers, M and N.
然后你要做的就是……
And what you do is…
计算a=m²-n²
a=m²-n².
此处假设M是较大的数
So we’re assuming M is the bigger one,
再求b=2MN
b=2MN,
以及c=M²+N²
and c=M²+N².
a b c就是一组毕氏三元数
And that is a Pythagorean triple.
只需选两个数 即可构造一组毕氏三元数
All you have to do is pick two numbers, and you can generate a Pythagorean triple.
毕式三元数有无穷多个
There’s an infinite number of them absolutely.
这是众所周知的 并不是什么秘密
So this is a well-known thing.This is not a mystery.
但问题是 能否用红色和蓝色给整数着色
But the mystery now is, can we color in the integers using red and blue,
使任一毕氏三元数不同色?
so that we don’t get a Pythagorean triple in the same color?
假设我有个蓝色的数集和一个红色的数集
So imagine me have a blue set numbers, and a red set of numbers.
红数集里不含任一组毕氏三元数
And the red set do not contain a Pythagorean triple,
蓝数集里也不含毕氏三元数
and the blue set do not contain a Pythagorean triple.
着色的方法固然很多
There’s lots of ways you can color your numbers.
但有没有一种能避免毕式三元数同色?
Is there’s a way to aviod this problem?
这就叫毕氏三元数问题
So this called the Pythagorean Triple Problem.
接下来就要用到拉姆齐定理了
Now this is an application of something called Ramsey theory.
拉姆齐定理探讨的是在大量对象中寻找某种结构
And Ramsey theory is about finding structure in large number of objects.
现有足够多的对象 某特定结构总会出现 这是必然的吗
So if you have a large number of objects, is a structure unavoidable, is it inevitable?
能避免出现同色的毕氏三元数吗?
Can you avoid this problem?
答案是 不能
Now, the answer is, no, you can’t avoid it.
这些人给出了为1到7824的整数着色
So what these guys have shown is that this is a solution
且保证毕氏三元数不同色的方法
that coloring the intergers from 1 to 7824.
他们把这些数分成了红数和蓝数
So they’ve split them up into red numbers blue numbers,
你应该可以看到图中的白色方块
and you see the white squares perhaps.
白方块里的数字可以是红色也可以是蓝色
The white squares represent numbers that could be red or blue.
这无所谓
And it doesn’t matter.
不管是红色还是蓝色都行得通
So it would be a solution either way.
但当他们再往前推一步时
But when they took it one step further,
即考虑从1到7825的整数的着色问题
and they looked at coloring in the integers from 1 to 7825.
他们就做不到确保毕氏三元数不同色了
That’s when it failed.
证明了这是不可能做到的
That’s when they showed that it can’t be done,
最终都会出现一组全红或全蓝的毕氏三元组数
that you’re always going to end up with a Pythagorean triple in the red set or in the blue set.
7825这个数正是压垮骆驼的最后一根稻草
This number 7825 is the straw that broke the camel’s back,
也是牛仔能装备上的最后一个物品
it’s the last item on buckaroo,
是它把之前的规律都打破了
it’s the thing that broke it all down.
7825同时存在于两组毕氏三元数中
The reason 7,825 broke the solution
所以此时毕氏三元数问题就无解了
is because it’s in two Pythagorean triples.
这两组数就是
here they are,
625²+7800²=7825²
625²+7800²=7825²
和5180²+5865²=7825²
and 5180²+5865²=7825²
所以他们发现的问题是
So what they found is that,
当他们在考虑7824的解时
when they looked at the solutions for 7824,
在任一可能的解中
you look at all the possible solutions,
625和7800总是同色的
625 and 7800 were always the same color,
我们就设它们都是蓝色吧
so they were either, let’s say they’re blue.
另外这两个数字5180和5865也总同色
And these numbers, 5180 and 5865,
但跟625和7800不同色
were always the same color and the other color,
此时我们可以说它们总是红色的
they were always red, perhaps, in this case,
这就意味着7825必须既红又蓝
which means 7825 now has to be red and blue at the same time,
这显然不可能 所以这个问题无解
which is not possible and the whole thing fails.
20世纪80年代 我们的朋友Ron Graham
And in the 1980s, our friend, Ron Graham, actually offered a prize
为将来能解决毕氏三元数问题的人
for the person who solved this problem,
设置了一笔奖金
the Pythagorean Triple Problem.
奖金设为100美元
He offered $100 prize,
现已转交给
which he is now delivered to
德克萨斯大学的一位计算机专家
one of the computer scientists at the University of Texas
他寄去了一张有效支票
He’s delivered the check.He’s paid up.
那么 要证明这是问题从1到7824的解
So to show that this is a solution for 1 to 7824,
或者说要证明或验证这是个可行解
to show that is a solution and to confirm it’s a solution,
花几秒在电脑上验证并不是很难
take seconds on a computer, it’s not very difficult to do.
但要证明这个问题
But to show there are no solutions,
在1到7825的整数上没有可行解
for 1 to 7825,
他就得检验每一种可能性
and he checked every possibility,
这要花很大的功夫来验算
would take a massive amount of computing time.
每个整数都有两种颜色可选
The number of ways you could fill those in,
那给所有整数着色就一共有
when each integer has two options,
2的7825次幂种着色方法
would be 2 to the power 7825.
这个数字太庞大了
And that number is so massive,
没错 你是可以找个超级计算机来算
Well, you could take a supercomputer,
但验算所需的时间仍然太久太久了
take too long for a supercomputer to check,
假设世上所有超级计算机
imagine all the supercomputers in the world,
一起来验算所有可能的着色方法
and imagine them checking all the possibilities
即便从宇宙大爆炸开始 从早算到晚
since the dawn of time, since the Big Bang,
还是算不完这么多种可能性
you still won’t be able to check all the possibilities.
所以他们没有选择这种笨办法
So that’s not what they did.
而是用上了一些数学技巧
What they did is they used some clever mathematical tricks
大大减少了必要的计算量
to reduce the number of things they had to check.
他们把这个问题的数量级降到了兆级
They boiled this down to about a trillion things that they had to check,
用德克萨斯大学的超级计算机来算
and that took them about 2 days
只用了两天左右就算完了
using a supercomputer in the University of Texas.
但这种证明方式唯一存在的问题是
The only problem really with this type of proof is
它并不会让我们更理解这为什么是正确的
it doesn’t increased our understanding of why this is true.
7825这个数字到底有什么特别之处呢?
What’s special about the number 7825?
为什么偏偏是这个数字呢?
Why that number?
这种依靠大量运算来完成的证明方式
These kind of proofs that require this huge amount of computation
并不会告诉我们要证的这个东西为什么正确
does not tell us anything about why something is true.
还有一种推测 无论我们用几种颜色来着色
And there is a conjecture that this will always be true
这个结论都是正确的
no matter how many colors we use.
我们可以用3种颜色来着色
We might use 3 colors,
也可以用4种颜色
or 4 colors,
则那个数字会越来越大
now that number is going to get larger and larger,
验算所需的计算量
and the amount of computation it takes to find where it fails
也会随之越来越大
is gonna be bigger and bigger.
但要证明这个结论恒成立
But to find a proof that shows it’s always true,
可能要回归到传统的数学方法来
that’s probably going to take traditional mathematics.
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The Great Courses Plus is an on-demand video learning service
由世界各地经验丰富的教师授课
with expert lecturers from all around the world,
内容涵盖了方方面面
covering all sorts of topics,
从核物理到罗马历史都有
from nuclear physics to Roman history.
当然你已经了解了数学类的课程
Now, of course that I’ve got you covered for mathematics,
如果你想深入了解当今局势
and this section here on trending and newsworthy topics is handy,
这里有世界趋势和新闻热点版块
if you want a deeper understanding of current events.
再来看看这个
And look at this,
训练爱犬101
Dog Training 101.
哇 Audrey 看来我们的周末有事做啦
That’s right, Audrey, it looks like our weekend is sorted.
是嘛 你也喜欢这套课程嘛?
Yeah? You’re up with some Dog Training 101?
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Now, you can join a free trial with access to all the videos,
当然也包括训狗课程
including the dog training,
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This is a great chance to dive deeper into the topics you love,
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will show them you came from Numberphile.
非常感谢“好课+”对本视频的支持
I’ll thanks to the Great Courses Plus for supporting this video.
– 谢谢- 谢谢
– Thank you.- Thank you.

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视频概述

关于毕达哥拉斯三元数问题的证明

听录译者

茄也

翻译译者

Mathilda

审核员

审核员_DB

视频来源

https://www.youtube.com/watch?v=1gBwexpG0IY

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