Pi hiding in prime regularities

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://youtu.be/DjI1NICfjOk
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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

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