Why Do Prime Numbers Make These Spirals? - Dirichlet’s Theorem, Pi Approximations, and More

Explore the fascinating world of prime numbers and their unexpected spiral patterns in this 22-minute video by 3Blue1Brown. Delve into the curious phenomenon of prime number spirals, uncover approximations for pi, and investigate prime distributions. Learn about residue classes, Euler's totient function, and Dirichlet's theorem as you journey through the mathematical landscape. Discover the connections between these concepts and their broader implications in number theory. Gain insights into the historical development of these ideas, including corrections to common misconceptions about Dirichlet's work on prime distribution. Engage with complex mathematical concepts through clear explanations and stunning visualizations, making abstract ideas more accessible and intriguing.

- The spiral mystery
- Non-prime spirals
- Residue classes
- Why the galactic spirals
- Euler’s totient function
- The larger scale
- Dirichlet’s theorem
- Why care?
: In the video, I say that Dirichlet showed that the primes are equally distributed among allowable residue classes, but this is not historically accurate. By "allowable", here, I mean a residue class whose elements are coprime to the modulus, as described in the video. What he actually showed is that the sum of the reciprocals of all primes in a given allowable residue class diverges, which proves that there are infinitely many primes in such a sequence.