Phase transition and computational complexity in a stochastic prime number generator

We introduce a prime number generator in the form of a stochastic algorithm. The character of such algorithm gives rise to a continuous phase transition which distinguishes a phase where the algorithm is able to reduce the whole system of numbers into primes and a phase where the system reaches a frozen state with low prime density. In this paper we firstly pretend to give a broad characterization of this phase transition, both in terms of analytical and numerical analysis. Critical exponents are calculated, and data collapse is provided. Further on we redefine the model as a search problem, fitting it in the hallmark of computational complexity theory. We suggest that the system belongs to the class NP. The computational cost is maximal around the threshold, as common in many algorithmic phase transitions, revealing the presence of an easy-hard-easy pattern. We finally relate the nature of the phase transition to an average-case classification of the problem.

💡 Research Summary

The paper introduces a stochastic algorithm that generates prime numbers by repeatedly selecting pairs of integers from a pool and applying a simple divisor‑removal rule: if one number divides the other, the divisor is eliminated and the quotient replaces the larger number. Starting with N random integers, the dynamics either drive the whole system toward a state where every element is prime (the “active” phase) or halt when no further divisor relationships exist, leaving a frozen configuration with a low density of primes.

Through extensive Monte‑Carlo simulations the authors identify a continuous phase transition separating these two regimes. They define the order parameter ρ(N, p) as the fraction of primes in the system, where p characterizes the initial size distribution of the numbers and N is the system size. As p is varied, ρ exhibits a sharp change at a critical value pc(N). Finite‑size scaling analysis yields the scaling form ρ(N, p)≈N−β/ν f