On a dynamical approach to some prime number sequences

TL;DR

This paper applies dynamical systems theory to analyze prime residue sequences, revealing their chaotic nature, non-uniform block distributions, and forbidden patterns, thereby offering a novel interdisciplinary perspective on prime number patterns.

Contribution

It introduces a dynamical systems approach to prime residue sequences, uncovering their chaotic properties and non-trivial entropy spectra, which is a new perspective in number theory research.

Findings

Prime residue sequences are maximally chaotic with non-uniform block distributions.

Prime gap residue sequences exhibit chaos but with forbidden patterns for larger blocks.

The spectrum of Renyi entropies is non-trivial, indicating complex underlying structure.

Abstract

In this paper we show how the cross-disciplinary transfer of techniques from Dynamical Systems Theory to Number Theory can be a fruitful avenue for research. We illustrate this idea by exploring from a nonlinear and symbolic dynamics viewpoint certain patterns emerging in some residue sequences generated from the prime number sequence. We show that the sequence formed by the residues of the primes modulo $k$ are maximally chaotic and, while lacking forbidden patterns, display a non-trivial spectrum of Renyi entropies which suggest that every block of size $m>1$, while admissible, occurs with different probability. This non-uniform distribution of blocks for $m>1$ contrasts Dirichlet's theorem that guarantees equiprobability for $m=1$. We then explore in a similar fashion the sequence of prime gap residues. This sequence is again chaotic (positivity of Kolmogorov-Sinai entropy), however chaos is weaker as we find forbidden patterns for every block of size $m>1$. We relate the onset of these forbidden patterns with the divisibility properties of integers, and estimate the densities of gap block residues via Hardy-Littlewood $k$-tuple conjecture. We use this estimation to argue that the amount of admissible blocks is non-uniformly distributed, what supports the fact that the spectrum of Renyi entropies is again non-trivial in this case. We complete our analysis by applying the Chaos Game to these symbolic sequences, and comparing the IFS attractors found for the experimental sequences with appropriate null models.