It comes in two parts. The first part is this. This is part one.

And part two is 200 terabytes large. That is huge!

I'm gonna start with something simpler, something perhaps you can try yourselves.

This is the idea. Let's take the numbers 1 to 9. I'm just gonna draw them out in a grid, here.

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

Now I'm interested in a+b=c.

These are gonna be whole numbers, and I"m gonna have a+b=c. That's not a difficult equation.

I want them to be different, so let's have a

Can I color these numbers, 1 to 9, using red and blue, so 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's no reason why not. And let's have 2 be a blue.

This means 3, it's 1+2... Well, that's okay. We can pick a color. It could be red or blue.

Brady, you can pick if you want. Do you want to be red?

Brady: Let's make it red.

Yeah, ok, fine. So we'll have 3 in red. Well, that's interesting because now we can make some deductions.

3 is red, 1 is red, so 4 can't be red. That's what we're 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.

1+3=4 would all be red. That would be something we're trying to avoid.

That means 4 is gonna be blue. Let's write 4 as blue. Great.

Well, that's something we can use as well. 2 is blue, 4 is blue, 6 will have to be red. Let's put 6 as red.

Well, 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, those are both red, so I'm gonna have 9 as blue. We're doing okay so far, don't have a problem.

Oh, but wait. There is a problem.

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 number has two options, red or blue, so it's gonna be 2 to the power 9. That's 512.

So there's 512 ways you could color this in.

It's not very hard for you to convince yourself that you're always going to fail, it's 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. A solution might be 3²+4². That is actually equal to 5².

Shall we just check and see if that's a solution? 3²=9, 4²=16. Add them together, 25. Great because 25=5².

Another solution to this might be 5²+12²=13². That is 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.

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 we have a blue set of 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.

Now, there's lots of ways you can color your numbers. Is there a way to avoid this problem?

This is called the Pythagorean Triple Problem.

Now, this is an application of something called Ramsey theory.

Ramsey theory is about finding structure in large numbers 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, coloring the integers from 1 to 7824.

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, 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.

In fact, you're always gonna end up with a Pythagorean triple in the red set or in the blue set.

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.

The reason 7825 broke the solution is because it's in two Pythagorean triples. Here they are.

625² + 7800² = 7825² and 5180² + 5865² is also equal to 7825²

So what they found is that when they looked at the solutions for 7824, you look at all the possible solutions,

625 and 7800 were always the same color. 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.

In the 1980s, our friend Ron Graham actually offered a prize for the person who solved this problem, the Pythagorean Triple Problem.

He offered a $100 prize, which he has 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, takes seconds on a computer. It's not very difficult to do.

But to show that 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, would be 2 to the power 7825.

And that number is so massive, well, you could take a supercomputer... it would 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 two days using a supercomputer in the University of Texas.

The only problem really with this type of proof is it doesn't increase our understanding of why this is true.

What is 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 three colors or four colors.

Now, that number's gonna 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 gonna take traditional mathematics.

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