A story of pi, primes, complex numbers, and how number theory braids them together. Mathologer on why 4k + 1primes break down as sums of squares: https://www.youtube.com/watch?v=DjI1NICfjOk Help fund future projects: https://www.patreon.com/3blue1brown An equally valuable form of support is to simply share some of the videos. Special thanks to these supporters: http://3b1b.co/leibniz-thanks Home page: https://www.3blue1brown.com/ For those of you curious about the finer details, here's a writeup from the viewer Daniel Flores justifying the final approximation: https://www.overleaf.com/read/wdzkfjbkwzyf The fact that only primes that are one above a multiple of four can be expressed as the sum of two squares is known as "Fermat's theorem on sums of two squares": https://goo.gl/EdhaN2 Music by Vince Rubinetti: https://vincerubinetti.bandcamp.com/album/the-music-of-3blue1brown Timestamps 0:00 - Introduction 1:39 - Counting lattice points 5:47 - Gaussian integers 10:30 - The lattice point recipe 17:50 - Counting on one ring 20:14 - Exploiting prime regularity 25:19 - Combining the rings 28:36 - Branches of number theory Thanks to these viewers for their contributions to translations Hebrew: Omer Tuchfeld ------------------ 3blue1brown is a channel about animating math, in all senses of the word animate. And you know the drill with YouTube, if you want to stay posted about new videos, subscribe, and click the bell to receive notifications (if you're into that). If you are new to this channel and want to see more, a good place to start is this playlist: http://3b1b.co/recommended Various social media stuffs: Website: https://www.3blue1brown.com/ Twitter: https://twitter.com/3Blue1Brown Patreon: https://patreon.com/3blue1brown Facebook: https://www.facebook.com/3blue1brown Reddit: https://www.reddit.com/r/3Blue1Brown