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• We talk about the two square theorem.

• 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?

• 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.

• 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.

We talk about the two square theorem.

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# The Prime Problem with a One Sentence Proof - Numberphile

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林宜悉 posted on 2020/04/05
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