On the Differences Between Consecutive Primes: Exploring the Prevalence of the Gap of 4

Prime numbers, the irreducible integers foundational to number theory, exhibit gaps whose distribution remains a subject of intense study. This paper investigates the recurring occurrence of a difference of 4 between consecutive primes, a phenomenon observed empirically but not yet fully understood theoretically. Using computational methods, we verify the prevalence of this gap for primes up to 105105, identifying 4,807 instances (10.1% of all gaps in this range). We contextualize these results within the Hardy-Littlewood conjectures, discuss computational limitations, and explore potential connections to the Riemann Hypothesis. Our findings highlight the interplay between empirical observation and theoretical conjecture in advancing the study of prime gaps.