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#### 7825问题

The Problem with 7825 - Numberphile

A new proof has been announced

which they claim is the largest proof ever.

It comes in two parts.

And the first part is this.

This is part 1.

And part 2 is 200 terabytes large.

That is huge!

something perhaps you can try yourselves.

Okay, this is the idea.Let’s take the numbers 1 to 9.

I’m just going to draw them out in a grid, here.

Imagine these are the numbers 1 to 9. I’m not gonna write the numbers in quite yet.

And now I’m interested ina+b=c.

So these are gonna be whole numbers, and I’m gonna have a+b=c.

That’s not a difficult equation.

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,

so that I don’t have a+b=c all the same color?

I don’t want them to be all red or all blue.That’s something I want to avoid.

So let’s try to see if we can.

Let’s start with 1 as a red.There is no reason why not.

And let’s have 2 be a blue.

This means, 3 is 1+2.

That’s okay. We can pick a color. It could be red or blue.
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.

So we’ll have 3 in red.

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.

That’s what we are trying to avoid.

We don’t want them all to be the same color,

because then we would have a+b=c all the same color,

or 1+3=4 would all be red.

That would be something we’re trying to avoid.

That means, 4, it’s gonna to be blue.

Let’s write 4 as blue, great!

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.

Let’s put 6 as red.

That’s interesting now. Because we’ve got 1+6=7.

We’re trying to avoid it being red.It’ll have to be blue.

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.

Okay, we’re doing okay so far, don’t have a problem…

Oh, but wait. There is a problem.
2+7＝9
Because 2+7=9.

And they’re all the same color.

Oh, no, I messed up.

I didn’t do it, I failed here, this has failed.

Well, there’s lots of other ways of coloring in the numbers 1 to 9,

In fact, each of them has two options, red or blue,

so it’s going to be 2 to the power 9, 512.

So there’s 512 ways you could color these in.

It’s not very hard for you to convince yourself that you’re always going to fail.

You’re always going to end up with something like that.

Or you could just check, all the options.

If you want, you could check the 512 options.

And you’ll see that it’s not possible

to avoid a+b=c in the same color.

That’s what this problem was about, but they took it one step further.

They were interested in a²+b²=c².

And they wanted to color in the numbers using red and blue

just like we did before.

But they wanted to avoid a²+b²=c² being the same color,

so all being red or all being blue.

Now you might recognize a²+b²=c².

That’s Pythagoras’s theorem.

But we’re interested in whole number solutions for this.

So a solution might be 3²+4².

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.

Great! Becase 25=5².
5²+12²＝13²也可以
Another solution to this might be 5²+12²=13².

That’s another solution that just uses whole numbers.

These are called Pythagorean triples.

They’re not particularly rare.In fact, the ancient Greeks knew how to make them.

I’ll show you how to make a Pythagorean triple, if you want.

You just take two numbers, M and N.

And what you do is…

a=m²-n².

So we’re assuming M is the bigger one,

b=2MN,

and c=M²+N².
a b c就是一组毕氏三元数
And that is a Pythagorean triple.

All you have to do is pick two numbers, and you can generate a Pythagorean triple.

There’s an infinite number of them absolutely.

So this is a well-known thing.This is not a mystery.

But the mystery now is, can we color in the integers using red and blue,

so that we don’t get a Pythagorean triple in the same color?

So imagine me have a blue set numbers, and a red set of numbers.

And the red set do not contain a Pythagorean triple,

and the blue set do not contain a Pythagorean triple.

There’s lots of ways you can color your numbers.

Is there’s a way to aviod this problem?

So this called the Pythagorean Triple Problem.

Now this is an application of something called Ramsey theory.

And Ramsey theory is about finding structure in large number of objects.

So if you have a large number of objects, is a structure unavoidable, is it inevitable?

Can you avoid this problem?

Now, the answer is, no, you can’t avoid it.

So what these guys have shown is that this is a solution

that coloring the intergers from 1 to 7824.

So they’ve split them up into red numbers blue numbers,

and you see the white squares perhaps.

The white squares represent numbers that could be red or blue.

And it doesn’t matter.

So it would be a solution either way.

But when they took it one step further,

and they looked at coloring in the integers from 1 to 7825.

That’s when it failed.

That’s when they showed that it can’t be done,

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,

it’s the last item on buckaroo,

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

is because it’s in two Pythagorean triples.

here they are,
625²+7800²=7825²
625²+7800²=7825²

and 5180²+5865²=7825²

So what they found is that,

when they looked at the solutions for 7824,

you look at all the possible solutions,
625和7800总是同色的
625 and 7800 were always the same color,

so they were either, let’s say they’re blue.

And these numbers, 5180 and 5865,

were always the same color and the other color,

they were always red, perhaps, in this case,

which means 7825 now has to be red and blue at the same time,

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

for the person who solved this problem,

the Pythagorean Triple Problem.

He offered \$100 prize,

which he is now delivered to

one of the computer scientists at the University of Texas

He’s delivered the check.He’s paid up.

So to show that this is a solution for 1 to 7824,

to show that is a solution and to confirm it’s a solution,

take seconds on a computer, it’s not very difficult to do.

But to show there are no solutions,

for 1 to 7825,

and he checked every possibility,

would take a massive amount of computing time.

The number of ways you could fill those in,

when each integer has two options,
2的7825次幂种着色方法
would be 2 to the power 7825.

And that number is so massive,

Well, you could take a supercomputer,

take too long for a supercomputer to check,

imagine all the supercomputers in the world,

and imagine them checking all the possibilities

since the dawn of time, since the Big Bang,

you still won’t be able to check all the possibilities.

So that’s not what they did.

What they did is they used some clever mathematical tricks

to reduce the number of things they had to check.

They boiled this down to about a trillion things that they had to check,

and that took them about 2 days

using a supercomputer in the University of Texas.

The only problem really with this type of proof is

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?

Why that number?

These kind of proofs that require this huge amount of computation

does not tell us anything about why something is true.

And there is a conjecture that this will always be true

no matter how many colors we use.

We might use 3 colors,

or 4 colors,

now that number is going to get larger and larger,

and the amount of computation it takes to find where it fails

is gonna be bigger and bigger.

But to find a proof that shows it’s always true,

that’s probably going to take traditional mathematics.
“好课+”是个可以点播视频学习的地方
The Great Courses Plus is an on-demand video learning service

with expert lecturers from all around the world,

covering all sorts of topics,

from nuclear physics to Roman history.

Now, of course that I’ve got you covered for mathematics,

and this section here on trending and newsworthy topics is handy,

if you want a deeper understanding of current events.

And look at this,

Dog Training 101.

That’s right, Audrey, it looks like our weekend is sorted.

Yeah? You’re up with some Dog Training 101?

Now, you can join a free trial with access to all the videos,

including the dog training,

by going to TheGreatCoursePlus.com/numberphile

This is a great chance to dive deeper into the topics you love,

and using the URL or the link in the video description

will show them you came from Numberphile.

I’ll thanks to the Great Courses Plus for supporting this video.
– 谢谢- 谢谢
– Thank you.- Thank you.

Mathilda