A Comprehensive Study of Prime Gaps Through Stochastic and Statistical Methods

Abstract This study delves into the stochastic behavior of prime gaps, which are the differences between consecutive prime numbers, across various samples with sizes ranging up to 100 billion consecutive prime gaps. By analyzing multiple models approximating the probability mass function (PMF) of prime gaps, we evaluated their performance using statistical tests such as the Chi-Square Test, Kolmogorov-Smirnov Test, Mean Squared Error (MSE), and Kullback-Leibler Divergence. A key finding is the significance of incorporating the modular behavior of prime gaps with respect to 6 (gn ≡ 0, 2, 4 (mod 6)) and adjusting the expectation of prime gaps in accurately predicting the approximated probability mass function (PMF) of prime gaps. Models that accounted for these modular properties and fine-tuned the expected value consistently outperformed those that did not. This research underscores the increasing complexity of expected prime gap behavior with larger n and suggests that prime gap distributions can be approximated by geometric distributions with parameter adjustments reflecting this complexity. The study’s findings have important implications for number theory and potential applications in cryptography. Future research directions include further exploration of modular properties, expanding the scope of statistical evaluations, and investigating additional mathematical properties influencing prime gap distributions.