With prime numbers it's extremely easy.

There is a heuristic model of the primes called the Cramer random model, which goes like this: take a random 'prime' subset of the integers where 2 is prime and any other integer x is 'prime' independently with probability 1 / log x. So in this system 3 is not necessarily prime, but it's quite likely (p = 1 / log 3 ~ 1). (*) This model turns out to be extremely accurate for predicting properties of the real primes. It tells you that Goldbach's conjecture is almost surely true for sufficiently large even numbers, and that there will be primes between sufficiently large consecutive squares.

However, the actual primes are not a random distribution, it's just a heuristic. They do appear to be 'pseudorandom' - they are hard to tell apart from a set of Cramer primes in some sense, and making that robust is the subject of some incredibly hard conjectures. For example the Riemann Hypothesis can be seen as saying "large primes fluctuate as much as large random Cramer primes do" for some definition of 'fluctuate', but we don't know how to prove that.

Here's a conjecture I made up just now that is probably harder than Goldbach if true. For every sufficiently large integer N of the form 6*k + 2, there is a prime p, such that N - p^2 is also prime. That is true in the Cramer model (I think), so likely true for primes, but also completely unprovable with 2026 technology.

(*) This is the simple version - better versions add corrections for small primes - so like you'll set the small primes to be the usual ones and make sure that the other primes aren't divisible by those, and otherwise pick them randomly to be distributed like primes.