Phase transition and computational complexity in a stochastic prime number generator

TL;DR

This paper studies a stochastic prime number generator, revealing a phase transition between prime reduction and frozen states, analyzing its critical behavior, and classifying its computational complexity as NP with an easy-hard-easy pattern.

Contribution

It characterizes the phase transition in a stochastic prime generator, computes critical exponents, and classifies the problem as NP, linking phase transition behavior to computational complexity.

Findings

Identifies a continuous phase transition in the prime generator.

Calculates critical exponents and demonstrates data collapse.

Shows the computational cost peaks at the threshold, indicating an NP classification.

Abstract

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.