At a high level it's not that far off, in the sense that most mathematicians think nontrivial patterns in the primes are at least unusual, and they mostly behav

At a high level it's not that far off, in the sense that most mathematicians think nontrivial patterns in the primes are at least unusual, and they mostly behave randomly. But it's true that there is some structure, which is in jargon terms called a "conspiracy" among the primes when it's found or hypothesized. As Terence Tao summarizes it, > We believe that the primes do not observe any significant pattern beyond the obvious ones (e.g. mostly being odd), but we are still a long way from making this belief completely rigorous. That's from this set of slides on structure and randomness in the primes, which has some other relevant bits in it: https://terrytao.files.wordpress.com/2009/07/primes1.pdf https://terrytao.files.wordpress.com/2009/07/primes1.pdf Especially relevant are slides 10-11 on treating the primes as a pseudorandom set, and then slides 14-15 on using pseudorandom models of the primes to rigorously (vs. heuristically) prove theorems. That's done by classifying and ruling out all possible ways nonrandom structure in the actual primes (the "conspiracies") could sink the specific theorem being proven.