  Natural numbers can be divided in two non-overlapping infinite sets, primes
and composites, with composites factorizing into primes. Despite their apparent
simplicity, the elucidation of the architecture of natural numbers with primes
as building blocks remains elusive. Here, we propose a new approach to decoding
the architecture of natural numbers based on complex networks and stochastic
processes theory. We introduce a parameter-free non-Markovian dynamical model
that naturally generates random primes and their relation with composite
numbers with remarkable accuracy. Our model satisfies the prime number theorem
as an emerging property and a refined version of Cram\'er's conjecture about
the statistics of gaps between consecutive primes that seems closer to reality
than the original Cram\'er's version. Regarding composites, the model helps us
to derive the prime factors counting function, giving the probability of
distinct prime factors for any integer. Probabilistic models like ours can help
to get deeper insights about primes and the complex architecture of natural
numbers.
