Is this the most beautiful proof? (Fermat's Two Squares)

A visual explanation of one-sentence proof of Fermat's Two Squares Theorem by Don Zagier.

If you're interested in this proof, I highly recommend it and Proofs From The Book! (In general, if you're interested in math, you should read Proofs From The Book)

Some Clarifications:
At 9:00, I meant to say 3^2 + 2^2. Sorry :p
When it comes to the Zagier Map, one question a friend asked is that we only ever consider cases where the inner square is the smallest square or the largest square - what about the in between cases? To this, try and think of if you can create a windmill with an "in-between" case. If you can/cannot, why? (spoiler you can't!)

The source code for the animations can be found here:

A huge thanks to Chaeyeon Lee, Youngjin Park, Oliver Ni, Eric Che and Antonio Kam for helping me review the video!

These animation in this video was made using 3blue1brown's library, manim:

Chapters:
0:00 Introduction
1:40 Part 1: The Involution
5:42 Part 2: The Windmill
10:47 Part 3: The Zagier Map
15:12 Putting it all together
17:13 Outro

Follow me!

Music (in order):
GameChops - National Park
GameChops - Route 113
Helynt - Route 47
Helynt - Bo-omb Battlefield
Helynt - Undewater
Helynt - Littleroot Town

Hope you guys enjoyed the music! I especially had fun picking out the tracks for this video. S/o to Helynt, they make amazing remixes!

The most beautiful proof in math? (Fermat's Two Squares Theorem)

Some tags: number theory, fermat's two squares, windmills, proofs, mathematics, vcubingx v cubingx, vcubing x

[vcubingx]

Hi all! It's been a while since I made a video. Was busy with work/school stuff, but now that that's wrapping up (hopefully!) I should be making more videos! I had a ton of fun making this one, especially picking out the song choices. Let me know what you think of this one!

[lowellrindler]

“My favorite thing about number theory is how simple the theorems are to state but how wildly complicated the proofs are.” This is also what I find so maddening about number theory, but to each their own haha.

[rahulshah]

At 8:56 i think "this tells us that 13 is 3^2 + 4^2" should've been 13 = 3^2 + 2^2 instead (in case anyone else was caught up by that), great video and incredible usage of graphics in a math proof!

[paulbloemen]

I am not a mathematician, but sometimes interested in patterns. I find this proof, and the subject itself, with all its aspects like windmills, totally mind boggling, I am so surprised that something like this even exists. Thank you for showing, and patiently explaining!

[artemetra]

omg. that's actually so elegant – one of those surprisingly common cases in math when a problem is extended to involve more variables and general statements in order to then fall back to the ground because some random pattern is seen.. these proofs are the ones that impress me the most, i have no clue how does one even come up with those things haha

also, you just gained a subscriber!

[wcremlj]

Fermat's Two Squares Theorem generally also includes the statement that the representation as a sum of two squares is actually *unique* for such primes. I don't think this proof can easily be extended to show this.

[aaronmartens]

Pretty slick, thank you. Counting things in two different ways is super powerful.

[djridoo]

Pokemon and Mario music on beautiful and elegant maths, what a cool video !

[yourgrandfather]

The notion of fixed point is so amazing.

For a set S, if we can prove that there is an involution f, such that f has odd number of its fixed points in S, then every involution on S has odd number of fixed points on S either.

Here we can prove that Zagier map is an involution and has (1, 1, k) as an only 1 fixed point on W_(p=4k+1z). Because 1 is odd, so all involution on W_(p=4k+1) including the flip map has at least 1 fixed point on W_(p=4k+1).

Since having a fixed point of the flip map on W_(p=4k+1) implies that p=4k+1 can be written as a sum of two squares, therfore we now have proved the theorem !

[jupa]

I'm to dumb for this but You've explained it beautifully. Well done. Who comes up with proofs like this? Insane.

[Rollinghills]

Here’s my one line proof. “This is left as an exercise for the reader.” 🔥🔥🔥

[shivbhatia]

This was an incredibly clear explanation, amazing work!

[chiaracoetzee]

This is a brilliant and really accessible explanation of this proof, to the point where it now seems almost self-evident, which is certainly not how I've ever felt about this theorem before! Incredible work, subscriber earned.

[varunmohanraj]

Omg the goat returns my favorite youtuber ❤️❤️❤️

[GrandMoffTarkinsTeaDispenser]

Congratulations on this beautiful video. A joy to watch from start to finish. This topic is very dear to me.

[mavericktron]

For a 4k+1 prime you start with the (1, 1, k) windmill put a pin in the middle and reel it in counterclockwise (aka involutions) then the middle gets fatter (Zair map) and you do it until you end up with four nice square flippers at the outside ... yah probably not exatly like that but that's how I saw it anyway 🙂

[-tsvk-]

The "windmill" construction is about warpping a 2d square with 2d rectangles on its all four sides, visualizing a mathemathical property/identity of the form a^2 + 4*b*c

Does there exist a corresponding construction in one dimension higher, wrapping a 3d cube with 3d cuboids on all of its six sides? Visualizing some possible related identity of the form a^3 + 6*b*c*d ?

[ingenuity-ygev]

oh my god. when you finally joined all the dots and went one by one through each line, it just all clicked and i felt really happy! amazing video!

[collingallaway]

While I can’t say I didn’t get lost in a few places, I still enjoyed the video and the concepts! Thanks for breaking the proof down into a comprehensive format

[justanormalyoutubeuser]

I just discovered your channel and it's absolutely fenomenal. You might have the best combination of rigour and intuition I've seen so far and the wildest thing is how you manage to make these things easy to follow. I'd love to see you cover something from algebraic topology, like number theory it is a subject where theorems are often easy to state but hard to prove (fixed point theorem, Borsuk Ulam theorem, sandwich theorem, hairy ball theorem, four color theorem, Euler's formula...).