Toward a dynamical model for prime numbers

Bonanno, Claudio; Mega, Mirko S.

doi:10.1016/s0960-0779(03)00433-8

Cited by 9 publications

(8 citation statements)

References 14 publications

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“…Littlewood gives the number of pairs of primes not necessarily successive and we would like to stress that in (2) τ d (x) denotes number of pairs of consecutive primes p n , p n+1 with difference p n+1 − p n = d. The pairs of primes separated by d = 2 and d = 4 are special as they always have to be consecutive primes (with the exception of the pair (3,7) containing 5 in the middle): in the triple of integers 2k+1, 2k+3, 2k+5 the middle 2k+3 has to be divisible by 3 if 2k + 1, 2k + 5 are prime (in particular not divisible by 3). For d = 6 (and larger d) we have π 6 (x) > τ 6 (x), for example (5,7,11), (7,11,13), (11,13,17), . .…”

Section: Nnsd For Prime Numbersmentioning

confidence: 99%

“…The expression (7) for τ d (x) was proved (in slightly different form required by the precision of the formulation of the theorem) under the assumption of the conjecture B of Hardy-Littlewood by D. A. Goldston and A. H. Ledoan [29] in 2012. During over a seven months long run of the computer program we have collected the values of τ d (x) up to x = 2 48 ≈ 2.8147 × 10 14 .…”

Section: 1])mentioning

confidence: 99%

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Nearest-neighbor-spacing distribution of prime numbers and quantum chaos

Wolf

2014

Phys. Rev. E

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We give heuristic arguments and computer results to support the hypothesis that, after appropriate rescaling, the statistics of spacings between adjacent prime numbers follows the Poisson distribution. The scaling transformation removes the oscillations in the nearest-neighbor-spacing distribution of primes. These oscillations have the very profound period of length six. We also calculate the spectral rigidity Δ(3) for prime numbers by two methods. After suitable averaging one of these methods gives the Poisson dependence Δ(3)(L) = L/15.

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Section: Nnsd For Prime Numbersmentioning

confidence: 99%

Section: 1])mentioning

confidence: 99%

Nearest-neighbor-spacing distribution of prime numbers and quantum chaos

Wolf

2014

Phys. Rev. E

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“…With the proper values of m and τ embedding parameters, a smooth discrete-time process is defined, which reconstructs the underlying dynamics. [2] The choice of these two parameters is crucial for the proper characterization of the structure of the time series [30, 31, 32]. The mathematical concepts for defining the state vector dimension m have been reviewed in details in Refs.…”

Section: Algorithm Design Using Symbolic Analysismentioning

confidence: 99%

Symbolic Information Flow Measurement (SIFM): A Software for Measurement of Information Flow Using Symbolic Analysis

Nebiu

Kamberaj

2019

Preprint

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AbstractSymbolic Information Flow Measurement software is used to compute the information flow between different components of a dynamical system or different dynamical systems using symbolic transfer entropy. Here, the time series represents the time evolution trajectory of a component of the dynamical system. Different methods are used to perform a symbolic analysis of the time series based on the coarse-graining approach by computing the so-called embedding parameters. Information flow is measured in terms of the so-called average symbolic transfer entropy and local symbolic transfer entropy. Besides, a new measure of mutual information is introduced based on the symbolic analysis, called symbolic mutual information.

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“…However, their global distribution exhibits an amazing regularity [10]. Certainly, this intriguing property contrasting local randomness and global evenness has caused the distribution of primes to be, since ancient times, a fascinating open problem for mathematicians [11] and more recently for physicists as well (see for instance [5,9,[12][13][14][15]). The Prime Number Theorem, which establishes the global smoothness of the counting function π(n) by providing the number of primes less or equal to the integer n, was the first indication of such regularity [16].…”

Section: A Statistical Pattern In the Prime Number Sequencementioning

confidence: 99%

“…12 Characteristic time τ as defined in the text versus N, for different pool sizes, from left to right: M = 2 10 , 2 11 , 2 12 , 2 13 , 214 . Every simulation is averaged over 2 × 10 4 realizations.…”

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confidence: 99%

Complex Systems, Numbers and Number Theory

Lacasa¹,

Luque²,

Miramontes³

2010

Reviews of Nonlinear Dynamics and Complexity

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All is number. PythagorasMuch ink has dried in the historical discussion of what is mathematics? Does it follows the scientific method? Is it just a collection of recipes, formulas and algorithms? Is it a science or a tool for others to use? Is it both? Carl Friedrich Gauss, one of the greatest mathematicians of all times called, it The Queen of Sciences but it was Eric Temple Bell who named it as Queen and Servant of Science in the title of one of his most celebrated books [1]. To some, this is an adequate way of expressing this dichotomy on the nature of mathematics and of its practical role ahead of other sciences [2]. There is, however, an aspect often neglected in all this discussion which is related to mathematics seen as natural phenomena and so susceptible of study by other sciences, such as physics. Do numbers along with their arithmetic operations behave as physical systems? More specifically, can the modern theory of nonlinear physics address number arithmetics as complex physical phenomena? From the remarkable coincidence in 1972 between H. Montgomery's work on the statistics of the spacings between zeta zeros and F. Dyson's analogous work on eigenvalues of random matrices, we have seen, somewhat unexpectedly, how number theory and physics have tended bridges between each other. These connections range from the reinterpretation of the Riemann zeta function as a partition function [3] or the focus of the Riemann Hypothesis via quantum chaos [4][5][6], to multifractality in the distribution of primes [7] or computational phase transitions in the number partitioning problem [8], to cite but a few (see [9] for an extensive bibliography).

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Toward a dynamical model for prime numbers

Abstract: We show one possible dynamical approach to the study of the distribution of prime numbers. Our approach is based on two complexity methods, the Computable Information Content and the Entropy Information Gain, looking for analogies between the prime numbers and intermittency.

Cited by 9 publications

References 14 publications

Nearest-neighbor-spacing distribution of prime numbers and quantum chaos

Nearest-neighbor-spacing distribution of prime numbers and quantum chaos

Symbolic Information Flow Measurement (SIFM): A Software for Measurement of Information Flow Using Symbolic Analysis

Complex Systems, Numbers and Number Theory

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