Title : Prime clustering and gaps via Pathak Continuum Compression Hypothesis (PCCH)
Abstract:
Prime numbers have always been of interest to mathematicians because of their seemingly random distribution pattern characterized by a tendency to cluster and then develop “gaps” between the primes. While traditional number theory has been able to statistically model and explain such phenomena as density and the asymptotic distribution of primes quite comprehensively, there hasn't been a “mechanistic” or “interaction”-based explanation for the pattern of primes clustering the way they do or for the pattern of the “gaps” developing between primes. In our study, we propose the Pathak Continuum Compression Hypothesis (PCCH). The PCCH models a number line, rather than an array of points in isolation, but a dynamic continuum which is subjected to forces of compression/elongation due to interactions between integers. In this model, each number has an interaction strength, denoted Interact, over its neighbouring numbers, which is a function of both proximity and similarity. This generates an array of points on a distorted numeric continuum, which may be modelled by a region of compression/elongation. The Pathak Compression Function is introduced to measure this. C(x) for integers up to 200 show remarkably regular patterns: negative compression scores are predominantly associated with prime numbers, while positive compression scores are associated with composite numbers. Regions of high negative compression are represented through clusters of primes, while areas of positive compression are easily observable through prime gaps. What is more significant is the fact that all variations of fundamental parameters of the model maintain this compression signature. The framework of PCCH also offers a clear rationale for known anomalies, including small primes and first composites, based on local asymmetry differences in interaction neighbourhoods for small primes and first composites. In summary, PCCH shifts the perspective of prime distribution from a GR or statistically based process to one that is a natural consequence of continuum deformation by numerical interaction. This paper proposes a paradigm shift in prime numbers, considering the prime number distribution as a resulting effect of interaction-related compression and expansion of the number continuum, and thereby unexplored avenues in prime numbers based on number theories.
