Does there exists other structures in math, which are seemingly random, but deterministic, and follow rules similar to the prime numbers, by rules I mean there must be statements similar to goldbach's conjecture or twin-prime-conjecture etc, for instance i dont consider the digits of pi to have this sort of structure. Is there some field in math which deals with distributions which "seem" random, but still follow certain rules ?
What are some examples of similar structures?
Also, I want to know what are the necessary axioms to produce the prime-number distribution, and if one ca replace these axioms to get other pseudo-random distributions which are not "isomorphic" to the prime-numbers, and which are not constructable in regular axiomatic systems ?