The set of primes: Towards an optimized algorithm, prime generation and validation, and asymptotic consequences

Starting from the result that the prime distribution is deterministic, we show in its maximal reduced form, the set of prime numbers P. This set can be expressed in terms of two subsets of N using three specific selection rules, acting on two sets of prime candidates. The asymptotic behaviour is considered too. We go head towards an optimized algorithm to generate primes, whose computational complexity is C(n)∈O(n). In addition a pre-computed algorithm is also considered and its computational complexity C(n)∈O(1).