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Entropy 2018, 20(2), 131; https://doi.org/10.3390/e20020131

On a Dynamical Approach to Some Prime Number Sequences

1
School of Mathematical Sciences, Queen Mary University of London, Mile End, London E1 4NS, UK
2
Escuela Tecnica Superior de Ingeniería Aeronáutica y del Espacio, Universidad Politécnica de Madrid, Madrid 28040, Spain
3
Departamento de Sistemas Complejos, Instituto de Física, Universidad Nacional Autónoma de México, CDMX 04510, Mexico
4
Centro de Ciencias de la Complejidad (C3), Universidad Nacional Autónoma de México, CDMX 04510, Mexico
*
Author to whom correspondence should be addressed.
Received: 8 December 2017 / Revised: 30 January 2018 / Accepted: 18 February 2018 / Published: 19 February 2018
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Abstract

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, unexpectedly 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. We numerically find that this sequence is again chaotic (positivity of Kolmogorov–Sinai entropy), however chaos is weaker as forbidden patterns emerge 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 Iterated Function System (IFS) attractors found for the experimental sequences with appropriate null models. View Full-Text
Keywords: entropy; symbolic dynamics; prime numbers; gap residues; chaos; fractals; complex systems; nonlinearity entropy; symbolic dynamics; prime numbers; gap residues; chaos; fractals; complex systems; nonlinearity
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This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited (CC BY 4.0).
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Lacasa, L.; Luque, B.; Gómez, I.; Miramontes, O. On a Dynamical Approach to Some Prime Number Sequences. Entropy 2018, 20, 131.

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