The problem with this visualization is that the pattern we see are meaningless.

It's making me thing of sieving, like in Sieve of Eratosthenes. But we have progressed a lot since.

What's nice about primes are the abstract ideas they generate, and the properties they have.

Primes connect numbers together. It allows you to form "groups" ( https://en.wikipedia.org/wiki/Group_(mathematics) ) of elements which don't have "subgroups". They are the sizes of the groups which don't have subgroups. (https://en.wikipedia.org/wiki/Cauchy%27s_theorem_(group_theo... )

If you want to know more about primes, you can look at the factorization of numbers problem.

You need to wrap your head around Euclid's algorithm (300 BC but it still matters a lot), this guy stumbled onto something very deep.

Prime numbers color other numbers. Once you pick a prime, number smaller that it can be colored as either "square" or "not square" (in the cyclic group defined by the prime). That's something figured Euler with https://en.wikipedia.org/wiki/Euler%27s_criterion .

It's one angle of attack to the factorization problem. This breaking of symmetry has been exploited in https://en.wikipedia.org/wiki/Quadratic_sieve .

But that's not the algorithm, I want to speak of today. I'd rather speak of https://en.wikipedia.org/wiki/Lenstra_elliptic-curve_factori... . The principle is simple, build a mathematical construct (elliptic curve) using the number which is not a prime and you want to factorize, and treat it as if it was a prime. Then you watch the construct crumble, trace the bug, and you have your factors.

If you are more mathematically proficient, you can also have a look at the General Number Field Sieve, but it's much harder.

To conclude I'd like to give a final nugget for the road : https://en.wikipedia.org/wiki/Montgomery_modular_multiplicat... a nice trick around division.