I’m not a mathematician so correct me if I’m wrong, but the patterns that emerge or more the natural result of the plotting method vs revealing anything meaningful about the distribution of primes.
> So what’s clear here is that the spirals themselves have nothing to do with prime numbers; a much cleaner and fuller pattern can be seen when we plot all positive integers (as well as zero).
Not a mathematician either but I think some quadratics have a tendency to produce more primes than others. What you are seing is the characteristic of various quadratics when plotted in this way. I plotted something quite simmilar.
If you're plotting primes, all the coordinates where you're not plotting are non-prime - so every 2nd coordinate will be blank. As will every 3rd and every 4th, 5th, 10th, 11th. etc etc.
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[ 2.8 ms ] story [ 24.4 ms ] threadhttps://en.wikipedia.org/wiki/Ulam_spiral
Why do prime numbers make these spirals? | Dirichlet’s theorem and pi approximations - 3Blue1Brown
https://www.youtube.com/watch?v=EK32jo7i5LQ
> So what’s clear here is that the spirals themselves have nothing to do with prime numbers; a much cleaner and fuller pattern can be seen when we plot all positive integers (as well as zero).
https://tessi.github.io/walking-the-ulam-spiral/
i built an "animation framework" in JavaScript around it where you can control and animate several parameters and even record the animation
https://primes.nickyreinert.de/
Surely that's where the pattern comes from.