Hidden structure in the randomness of the prime number sequence?

TL;DR

This paper presents a rigorous theory explaining the periodic patterns in prime number differences, linking number theory with statistical mechanics and fractal geometry, supported by numerical validation.

Contribution

It introduces a novel theoretical framework that explains periodic behaviors in prime differences and connects them to multiple mathematical and physical concepts.

Findings

Identifies the origin of periodic patterns in prime differences.

Links prime number behavior to spin systems and fractals.

Provides numerical evidence supporting the theory.

Abstract

We report a rigorous theory to show the origin of the unexpected periodic behavior seen in the consecutive differences between prime numbers. We also check numerically our findings to ensure that they hold for finite sequences of primes, that would eventually appear in applications. Finally, our theory allows us to link with three different but important topics: the Hardy-Littlewood conjecture, the statistical mechanics of spin systems, and the celebrated Sierpinski fractal.