This concerns prime numbers.

You know, the numbers which are only divisible by 1 and themselves.

The question: how to write a prime number as the sum of two squares of natural numbers

- of positive natural numbers.

The easiest example is: take the prime number 2, the smallest prime number.

You can write this as 1 squared plus 1 squared.

Let's try the next prime, 3.

Well... 1 squared plus 1 squared - obviously not.

1 squared plus 2 squared - already too big, so, no.

Argh.

5, 1 squared plus 2 squared. Works.

Let's do this

Fourth example: 7.

You try 1 squared plus...

6, 6 is not a square...

2 squared is 4, plus 3, 3 is not a square.

Uh, we see it's not possible.

And the question is:

Do there exist natural numbers,

x and y,

such that p is x squared plus y squared.

We're asking for a simple criterion

which for an arbitrary prime number

we know that there are arbitrarily large prime numbers, say,

with 10 million digits.

You tell me or I tell you

can be written as the sum of 2 squares

or not.

And if the prime number is so large

we cannot try it out like we did with the small primes

Brady: "Are we looking here for a formula, or, like, a way to get those squares, or just...?"

No. Impossible. That would be great.

Nobody knows a formula.

You, you can only decide if you have good luck

whether it is possible at all, not a formula

which gives you x and y.

If you would give me some prime number, say you search on the internet

for large prime numbers,

you give me one and then tell professor, now tell me yes or no,

and then, within a second I will tell you yes or no.

Brady: "But you won't be able to tell me what those squares are."

Absolutely not. Nobody will be able. Not only me.

I suggest that we look into the history briefly.

Who did this first? This is the famous French mathematician

Pierre de Fermat, who did other important things, and he liked such easy to formulate problems

And what did he do with these problems?

He made tests. He made long tables.

So he took, we have here four primes,

he took the primes up to hundred,

the primes up to thousand,

wrote down this list

and checked by hand, case by case.

If you do that, we can, we should show such a list,

at the first glance you don't see any pattern.

But Fermat finally observed a very very simple rule.

And that's already his theorem which I write down now.

Prime number p is the sum of two squares if and only if

we subtract 1 from p, get a nice number, if this number is divisible by 4.

That means you can make a very very simple test. So let's consider small prime.

Let's say 17.

We subtract one, we get 16. Ah!

Divisible by four? It works. And now we check, is it really true?

And we see, ah, 17 is 4 times 4 plus 1 times 1, true.

Let's consider 23, other nice prime. We subtract 1, get 22, not divisible by 4

So, I, you can try it yourself at home, you will not be able to write it as the sum of two squares.

And the remarkable thing is this is an easy test to subtract, and to check whether it's divisible by 4.

It's very easy, and you can try now on the internet, write down huge prime numbers

subtract 1, try to divide by 4, and you know, here you go, and you will not be able to find x and y.

He saw it and he tested it again and again.

We don't know whether he wrote down a proof.

He said it's a theorem.

But he is famous for this.

He likes, when he was convinced that something is true, he probably believed in God

and in the order in the world, and he said,

if I see this so often, it must be true.

So he wrote down theorem, but without proof.

And the mathematicians after him wondered about the proof. Very good mathematicians, and could not do it.

Hundred years later, the Swiss mathematician Euler gave the first proof.

A tricky proof.

Complicated. Too complicated for this video. First proof.

Then, fifty years later, Gauss gave a completely different proof.

Seventy years later, Dedekind gave again a completely different proof.

So the mathematicians like to give different proofs of the same theorem because of

every proof sheds light on the statement,

and all these proofs shed different light on the statement.

So that's, all these proofs are too complicated even to speak about that here.

But about fifteen years ago, the mathematician Don Zagier from Bonn gave the famous one sentence proof.

And that we are going to talk about.

Brady: "One sentence?!"

In one sentence. Of course you will see for an ingenious mathematician like Zagier,

it can be done in one sentence.

For ordinary mathematicians like me, I need already ten sentences.

And for ordinary people we will add a few more.

...numbers in particular positive numbers. Because of if you would have infinitely many solutions

then these numbers would get arbitrarily large.

And p is a fixed number, so we would get on the right side numbers which are larger than p.