This is a paper I wrote some time ago, where I demonstrate, using the simplest operations, how to correctly generate prime numbers. The algorithms are extremely inefficient and only serve the purpose of showing a new way of thinking about prime numbers. I'd like to demonstrate that, contrary to current belief, the distribution of primes are not choatic beyond comprehensible rules or laws. I timestamped the paper using the Bitcoin blockchain to ensure originality of ideas.
I should emphasize, before you automatically assume, that this method is NOT another sieve (and does NOT rely on one), at least in the sense of filtering out composites by exhaustive elimination relatively to potential factors (like the Sieve of Eratosthenes). In fact, it is quite the opposite. Anybody who has programmed the Sieve of Eratosthenes should notice the radically different approach; I do not rely on composites. If I had to compare, I would say that in essence it is a bit similar to the chaos game of the Sierpiński triangle, where primes are analogous to the computed points that draws the (ever-denser) triangles in the pattern, while the composites are anologous to the (empty) triangular areas we never reach while playing the game to "infinity's end."
I have not found anything in the current state of research in number theory that aligns with my approach. If you think there is, please point me in the right direction.