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Natural numbers can be divided in two non-overlapping infinite sets, primesand composites, with composites factorizing into primes. Despite their apparentsimplicity, the elucidation of the architecture of natural numbers with primesas building blocks remains elusive. Here, we propose a new approach to decodingthe architecture of natural numbers based on complex networks and stochasticprocesses theory. We introduce a parameter-free non-Markovian dynamical modelthat naturally generates random primes and their relation with compositenumbers with remarkable accuracy. Our model satisfies the prime number theoremas an emerging property and a refined version of Cram\'er's conjecture aboutthe statistics of gaps between consecutive primes that seems closer to realitythan the original Cram\'er's version. Regarding composites, the model helps usto derive the prime factors counting function, giving the probability ofdistinct prime factors for any integer. Probabilistic models like ours can helpto get deeper insights about primes and the complex architecture of naturalnumbers.

The complex architecture of primes and natural numbers