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


From Mathologer.

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.

I learned about the new visual proof from someone who goes by the YouTube name TheOneThreeSeven. What TheOneThreeSeven pointed out to me was a summary of the windmill proof by Moritz Firsching in this mathoverflow discussion: https://mathoverflow.net/questions/31113/zagiers-one-sentence-proof-of-a-theorem-of-fermat
In turn Moritz Firsching mentions that he learned this proof from Günter Zieger and he links to a very nice survey of proofs of Fermat’s theorem by Alexander Spivak that also contains the new proof (in Russian): Крылатые квадраты (Winged squares), Lecture notes for the mathematical circle at Moscow State University, 15th lecture 2007: http://mmmf.msu.ru/lect/spivak/summa_sq.pdf
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"

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:
e^i pi=-1 : https://youtu.be/-dhHrg-KbJ0 (there are actually a couple of videos in which I talk about this but this is the main one)
infinitely many primes: Mentioned a couple of times: This video has a really fun proof off the beaten track:https://youtu.be/LFwSIdLSosI
pi^2/6: Again mentioned a couple of times but this one here is the main video: https://youtu.be/yPl64xi_ZZA
root 2 is irrational: one of the videos in which I present a proof: https://youtu.be/f1yDExNAEMg
pi is transcendental: https://youtu.be/9gk_8mQuerg
And actually there is one more on the list, Brouwer’s fixed-point theorem that is a corollary of of what I do in this video: https://youtu.be/7s-YM-kcKME
– When you start with the 11k windmill and then alternate swapping yz and the footprint construction, you’ll start cycling through different windmill solutions and will eventually reach one of the solutions we are really interested in. Zagier et al talk about this in an article in the American Mathematical Monthly "New Looks at Old Number Theory" https://www.jstor.org/stable/10.4169/amer.math.monthly.120.03.243?seq=1

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