Why was this visual proof missed for 400 years? (Fermat's two square theorem)

Today's video is about a new really wonderfully simple and visual proof of Fermat's famous two square theorem: An odd prime can be written as the sum of two integer squares iff it is of the form 4k+1. This proof is a visual incarnation of Zagier's (in)famous one-sentence proof.

0:00 Intro
2:20 Chapter 1: Discovering a theorem
7:05 Chapter 2: 400 years worth of proofs
9:59 Chapter 3: Zagier's one-sentence proof
15:40 Chapter 4: The windmill trick
22:12 Chapter 5: Windmill maths interlude
25:08 Chapter 6: Uniqueness !!
33:08 Credits

The first ten minutes of the video are an introduction to the theorem and its history. The presentation of the new proof runs from 10:00 to 21:00. Later on I also present a proof that there is only one way to write 4k+1 primes as the sum of two squares of positive integers.

Here is a link to JSTOR where you can read Zagier's paper for free:
Here are the Numberphile videos on Zagier's proof that I mention in my video:

Finally here is a link to my summary of the different cases for the windmill pairing that need to be considered (don't read until you've given this a go yourself :)

Today's t-shirt is one of my own: "To infinity and beyond"
Enjoy!

P.S.: Added a couple of hours after the video went live:
One of the things that I find really rewarding about making these videos is all the great feedback here in the comments. Here are a few of the most noteworthy observations so far:
- Based on feedback by one of you it looks like it was the Russian math teacher and math olympiad coach Alexander Spivak discovered the windmill interpretation of Zagier's proof; see also the link in the description of this video.
- Challenge 1 at the very end should (of course :) be: an integer can be written as a difference of two squares if and only if it is odd or a multiple of 4.
-one of you actually ran some primality testing to make sure that that 100 digit number is really a prime. Based on those tests it's looking good that this is indeed the case :)
- one of you actually found this !!! 6513516734600035718300327211250928237178281758494417357560086828416863929270451437126021949850746381 = 16120430216983125661219096041413890639183535175875^2 + 79080013051462081144097259373611263341866969255266^2
- a nice insight about the windmill proof for Pythagoras's theorem is that you can shift the two tilings with respect to each other and you get different dissection proofs this way. Particularly nice ones result when you place the vertices of the large square at the centres of the smaller squares :)
- proving that there is only one straight square cross: observe that the five pieces of the cross can be lined up into a long rectangle with short side is x. Since the area of the rectangle is the prime p, x has to be 1. Very pretty :)
- Mathologer videos covering the ticked beautiful proofs in the math beauty pageant:

[caspermadlener]

This proof is so beautiful that I wrote an entire essay about numbers as the sum of two squares. When the essay was "finished" (I admit that it wasn't), I sent it to the main competition for this type of math essays in the Netherlands, and it got third place.
Also, because I heavily studied the subject in my spare time and Olympiad training, I got really good at this type of number theory.
When I participated at the IMO in Oslo this year (second time), I solved question 3 with full points, which was about this type of number theory. I got a perfect score on the first day, and scored 7+5+4=16 points on the second day, for a total of 37 points! GOLD! 19th place worldwide! Relative best for my country ever!
I really don't know if I would have gotten this score without this proof, so thank you so much for making this video. I hope that you are going to inspire lots of other people as well!

[amirilan]

Wow, the windmill proof is such a beautiful proof, amazing.

[DudeBoerGaming]

I have done a so much work on Z=X²-Y² it nearly drove me insane! Spent like 2 years on it with the hopes of making progress in solving the prime factorization challenge. Anyhow...

[DonnieTheKing]

Challenge: If n is odd, say n = 2k + 1, then
(k + 1)^2 - k^2 = k^2 + 2k + 1 - k^2 = n.


More generally, a nonzero difference of squares has to look like x^2 - y^2, where x > y,
say x = y + d. Then
x^2 - y^2 = (y + d)^2 - y^2 = 2d + d^2.
If d is odd then this gives an odd number, and if d is even then this gives a multiple of 4.


Using the above, if p is prime and p = 2d + d^2 it's clear that d must equal 1, which means the only way to express p is (y + 1)^2 - y^2.

[denisdaly]

I loved this video. I was able to follow it, and learned as well. Very interesting.

[martinepstein]

Amazing video as always!
I see some commenters sharing their favorite theorems. In the theme of counting how many objects can be created in a certain way I recently learned about Kurotowski's closure-complement problem. It asks: given any subset of any topological space, by taking successive closures and complements how many different sets can be created? The answer turns out to be 14 ! What a strange number. It seems too high, but if you smush together enough weird subsets of R you can achieve it.

[elonmusk]

The remaining primes which are not in the form of 4k+1can be written as 4k-1or 4k-3. It is easier to write in the form of 4k-1.

[amrutajoshi]

Yes that number can be written in form 4k+1, because the given number - 1 is divisible by 4

[abhinavshripad]

After seeing this proof it looks like that this theorem could have been put in an Olympiad problem also

[Achill]

Comment just for myself as reminder why 4k+3 is not sum of two squares for any k:
Assume there are m and n with
m^2 + n^2 = 4k + 3 : The right side is odd, so either m or n is odd.
Assume n is even and m is odd: n = 2p and m = 2q + 1 then
m^2 + n^2 = 4p^2 + 4q^2 + 4q + 1 = 4k +3
The left side is 1 (mod 4) and the right side is 3 (mod 4), which is impossible.

Therefore, no such n and m can exist.

[rishijai]

You know you are a maths Mafia boss when you make a video to correct numberphile 😁

[MrJesuisanonyme]

If a prime number is the sum of two squares, it has to be of the form 4 k + 1.

Well, obviously.

As you showed, one square must be odd, and the other even, and their roots too.

Let's write the odd one as (2 x + 1)², and the even one as (2 y)².

Then p = (2 x + 1)² + (2 y)²

p = 4 x² + 4 x + 1 + 4 y²

p = 4 (x² + x + y²) + 1

What did I miss?

[scooldrood]

"4k+1, now can you see the patter on the left?"
"Yeah 😄, 4k-1!"
"4k+3!
"😑"

[FourthDerivative]

"The proof is left as an exercise for the reader" -Fermat

[raynmanshorts]

Fermat: "Hey, here's this cool thing about numbers."
Mathematicians: "Amazing! Can you prove it?"
Fermat: "I already did."
Mathematicians: "Wow! Can we see it?"
Fermat: "Hmmm... nah."

[mikemthify]

This proof was discovered by Roger Heath-Brown in 1971, and was later condensed into the one sentence version by Don Zagier. It's one of two proofs of this theorem found in the wonderful book "Proofs from THE BOOK" 6th ed by Martin Aigner and Günter M. Ziegler in chapter 4.

[TheOneThreeSeven]

A year ago I left a comment on one of these video's saying I was so inspired I was going to make my own math education you tube video's. I have something very special for everyone coming very soon, it's a free software project that I created while working on a tool to make animations for my video's and is almost ready to be released. I just published the first video on my channel, check it out!

[Mathologer]

In his 1940 book “A Mathematician’s apology” the mathematical superstar G.H. Hardy writes: “Another famous and beautiful theorem is Fermat’s ‘two square’ theorem... All the primes of the first class” [i.e. 1 mod 4] ... “can be expressed as the sum of two integral squares... This is Fermat’s theorem, which is ranked, very justly, as one of the finest of arithmetic. Unfortunately, there is no proof within the comprehension of anybody but a fairly expert mathematician.”
My mission in today’s video is to present to you a beautiful visual proof of Fermat’s theorem that hardly anybody seems to know about, a proof that I think just about anybody should be able to appreciate. Fingers crossed :) Please let me know how well this proof worked for you.
And here is a very nice song that goes well with today’s video:


Added a couple of hours after the video went live:
One of the things that I find really rewarding about making these videos is all the great feedback here in the comments. Here are a few of the most noteworthy observations so far:

-Based on feedback by one of you it looks like it was the Russian math teacher and math olympiad coach Alexander Spivak discovered the windmill interpretation of Zagier's proof; see also the link in the description of this video.
-Challenge 1 at the very end should be (of course :) be: an integer can be written as a difference of two squares if and only if it is odd or a multiple of 4.
-one of you actually some primality testing to make sure that that 100 digit number is really a prime. Based on those tests it's looking good that this is indeed the case :)
-one of you actually found this !!! = +
- a nice insight about the windmill proof for Pythagoras's theorem is that you can shift the two tilings with respect to each other and you get different dissection proofs this way. Particularly nice ones result when you place the vertices of the large square at the centres of the smaller squares :)
-proving that there is only one straight square cross: observe that the five pieces of the cross can be lined up into a long rectangles one of whose short side is x. Since the area of the rectangle is the prime p, x has to be 1. Very pretty :)
-Mathologer videos covering the various ticked beautiful theorems:

[serkanmuhcu]

I like, that 3blue1brown is also a patron

[jakegerke]

I never made it past geometry in public school, and yet I was able to follow most of this well, and appreciate how beautiful this proof really is. I chalk that up not only to your ability to explain things in various ways, but also to just how clean and professionally edited this video was. Well done. You have yourself a new fan. (Or... a new windmill.)