Mathematica Eterna
Open Access
ISSN: 1314-3344
+44-77-2385-9429
ISSN: 1314-3344
+44-77-2385-9429
This paper presents a comprehensive stochastic framework for modeling the Probability Mass Function (PMF) of prime gaps, focusing on the existence and characterization of a stationary distribution. Through theoretical analysis, we establish that prime gap sequences can be effectively modelled as stochastic processes and provide a formal proof of the existence of a stationary distribution that governs their long-term behavior.
To support our theoretical findings, we conducted an extensive numerical analysis using the Canadian supercomputer "Béluga", computing prime gaps up to 1012 using the Sieve of Eratosthenes. The empirical results validate our models, particularly those incorporating arithmetic properties such as prime factorization and modulo 6 congruence, demonstrating that they capture the periodic and combinatorial nature of prime gap distributions more accurately than simpler models.
Our study reveals that approximations adopting a geometric PMF structure with piecewise components outperform non-piecewise models and align with the stationary distribution. Specifically, prime gaps of 2 and 4 exhibit a uniform distribution, comprising approximately 5% of all gaps, while larger gaps follow a geometric progression. This dual nature of prime gaps-uniform for smaller gaps and structured for larger ones-offers a novel perspective on their distribution. Our formal proof of the stationary distribution not only enhances the understanding of prime gap distributions but also contributes to the broader field of stochastic modeling in prime number theory.
Published Date: 2024-09-23; Received Date: 2024-08-20