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Grant support

Lucas Lacasa acknowledges funding from an EPSRC Early Career Fellowship EP/P01660X/1. Octavio Miramontes acknowledges funding from PASPA-UNAM for a sabbatical grant and the PAPIT-UNAM research grant IN105015. Octavio Miramontes is very grateful with the Technical University of Madrid in Spain for hosting a sabbatical leave.

Analysis of institutional authors

Luque, BartolomeAuthorGomez, IgnacioAuthor

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June 9, 2019
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Article

On a Dynamical Approach to Some Prime Number Sequences

Publicated to:Entropy. 20 (2): E131- - 2018-02-01 20(2), DOI: 10.3390/e20020131

Authors: Lacasa, Lucas; Luque, Bartolome; Gomez, Ignacio; Miramontes, Octavio;

Affiliations

Queen Mary Univ London, Sch Math Sci, London E1 4NS, England - Author
Univ Nacl Autonoma Mexico, Ctr Ciencias Complejidad C3, Mexico City 04510, DF, Mexico - Author
Univ Nacl Autonoma Mexico, Dept Sistemas Complejos, Inst Fis, Mexico City 04510, DF, Mexico - Author
Univ Politecn Madrid, Escuela Tecn Super Ingn Aeronaut & Espacio, E-28040 Madrid, Spain - Author

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.

Keywords

ChaosComplex systemsEntropyFractalsGap residuesNonlinearityPrime numbersSymbolic dynamics

Quality index

Bibliometric impact. Analysis of the contribution and dissemination channel

The work has been published in the journal Entropy due to its progression and the good impact it has achieved in recent years, according to the agency WoS (JCR), it has become a reference in its field. In the year of publication of the work, 2018, it was in position 28/81, thus managing to position itself as a Q2 (Segundo Cuartil), in the category Physics, Multidisciplinary. Notably, the journal is positioned en el Cuartil Q2 para la agencia Scopus (SJR) en la categoría Physics and Astronomy (Miscellaneous).

From a relative perspective, and based on the normalized impact indicator calculated from the Field Citation Ratio (FCR) of the Dimensions source, it yields a value of: 4.33, which indicates that, compared to works in the same discipline and in the same year of publication, it ranks as a work cited above average. (source consulted: Dimensions Jul 2025)

Specifically, and according to different indexing agencies, this work has accumulated citations as of 2025-07-16, the following number of citations:

  • WoS: 2
  • Scopus: 3

Impact and social visibility

From the perspective of influence or social adoption, and based on metrics associated with mentions and interactions provided by agencies specializing in calculating the so-called "Alternative or Social Metrics," we can highlight as of 2025-07-16:

  • The use, from an academic perspective evidenced by the Altmetric agency indicator referring to aggregations made by the personal bibliographic manager Mendeley, gives us a total of: 8.
  • The use of this contribution in bookmarks, code forks, additions to favorite lists for recurrent reading, as well as general views, indicates that someone is using the publication as a basis for their current work. This may be a notable indicator of future more formal and academic citations. This claim is supported by the result of the "Capture" indicator, which yields a total of: 9 (PlumX).

With a more dissemination-oriented intent and targeting more general audiences, we can observe other more global scores such as:

  • The Total Score from Altmetric: 1.
  • The number of mentions on the social network X (formerly Twitter): 3 (Altmetric).

It is essential to present evidence supporting full alignment with institutional principles and guidelines on Open Science and the Conservation and Dissemination of Intellectual Heritage. A clear example of this is:

  • The work has been submitted to a journal whose editorial policy allows open Open Access publication.
  • Assignment of a Handle/URN as an identifier within the deposit in the Institutional Repository: https://oa.upm.es/86430/

As a result of the publication of the work in the institutional repository, statistical usage data has been obtained that reflects its impact. In terms of dissemination, we can state that, as of

  • Views: 42
  • Downloads: 134

Leadership analysis of institutional authors

This work has been carried out with international collaboration, specifically with researchers from: Mexico; United Kingdom.

There is a significant leadership presence as some of the institution’s authors appear as the first or last signer, detailed as follows: Last Author (Miramontes, Octavio).

the author responsible for correspondence tasks has been Miramontes, Octavio.