The Race
This video sets out a race between two sets of primes - just like we covered in Chebyshev's Bias:
- Primes $\equiv 1 \pmod{4}$ - primes that leave a remainder of 1 when divided by 4 (e.g., 5, 13, 17, 29...)
- Primes $\equiv 3 \pmod{4}$ - primes that leave a remainder of 3 when divided by 4 (e.g., 3, 7, 11, 19...)
Watch the Video
Key Discoveries
The main discovery here is not so much that there is a bias, but that $3 \pmod{4}$ leads the race 99% of the time.
The weird thing is that $1 \pmod{4}$ will always have a chance of being in the lead again at any point, however big the numbers get.
Something is bringing the numbers back together. The contention of this whole site: it is the Riemann Zeta Function.
Great video, great explanation by Numberphile featuring 3Blue1Brown!