A visual explanation of one-sentence proof of Fermat's Two Squares Theorem by Don Zagier. Support me on Patreon! This video adapted a lot of material from this paper: 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! Website: Twitter: Github: Instagram: Patreon: Music (in order): Philanthrope, mommy - embrace GameChops - National Park Philanthrope, Idealism - Still GameChops - Route 113 Helynt - Route 47 Helynt - Bo-omb Battlefield Helynt - Undewater Knowmadic - Faces 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