Toward a dynamical model for prime numbers

2004

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.

2003

The difference between two consecutive prime numbers is called the distance between the primes.

Entropy

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.

2014

Natural numbers can be divided in two non-overlapping infinite sets, primes and composites, with composites factorizing into primes. Despite their apparent simplicity, the elucidation of the architecture of natural numbers with primes as building blocks remains elusive. Here, we propose a new approach to decoding the architecture of natural numbers based on complex networks and stochastic processes theory. We introduce a parameter-free non-Markovian dynamical model that naturally generates random primes and their relation with composite numbers with remarkable accuracy. Our model satisfies the prime number theorem as an emerging property and a refined version of Cram\'er's conjecture about the statistics of gaps between consecutive primes that seems closer to reality than the original Cram\'er's version. Regarding composites, the model helps us to derive the prime factors counting function, giving the probability of distinct prime factors for any integer. Probabilist...

Print ISBN: 978-93-48388-06-3, eBook ISBN: 978-93-48388-54-4, 2024

In this paper I present the distribution of prime numbers which was treated in many researches by studying the function of Riemann; because it has a remarkable property; its non trivial zeros are prime numbers; but in this work I will show that we can find the distribution of prime numbers on remaining in natural numbers onl

International Journal of Innovative Technology and Exploring Engineering, 2019

In this paper we probed some interesting aspects of primorial and factorial primes. We did some numerical analysis about the distribution of prime numbers and tabulated our findings. Also, we pointed out certain interesting facts about the utility value of the study of prime numbers and their distributions in control engineering and Brain networks.

We prove that the digits of the primes are aperiodic in all bases with a single exception. We introduce a set of related theorems that regulate the behaviour of the natural numbers through the notion of periodicity and the computational mechanism of the binary derivative. We use these theorems to establish and then investigate the behaviour of a metric p(s') which is an analytic probability of primality. This metric is based purely upon the periodicity observed in a binary number and its binary derivatives. We demonstrate that this metric is exactly quadratic. We empirically discover a small stochastic imbalance in the number of primes in the two halves of the natural numbers partitioned by their final binary derivative. We show that this stochastic imbalance must vanish in the limit such that the variance of the difference between Pi(x) and Li(x) tends to zero. This confirms our earlier work via a different method. Proof of the Riemann Hypothesis implicitly follows through the 1901 equivalence of Von Koch. We again use our metric to reorder the number line and show that the related prime density is quadratic.

2017

Currently there is no known efficient formula for primes. Besides that, prime numbers have great importance in e.g., information technology such as public-key cryptography, and their position and possible or impossible functional generation among the natural numbers is an ancient dilemma. The properties of the functions 2ab+a+b in the domain of natural numbers are introduced, analyzed, and exhibited to illustrate how these single out all the prime numbers from the full set of odd numbers. The characterization of odd primes vs. odd non-primes can be done with 2ab+a+b among the odd natural numbers as an analogue to the other, well known type of fundamental characterization for irrational and rational numbers among the real numbers. The prime number theorem, twin primes and erratic nature of primes, are also commented upon with respect to selection, as well as with the Fermat and Euler numbers as examples.

RATIO MATHEMATICA, 2019

Within the conceptual framework of number theory, we consider prime numbers and the classic still unsolved problem to find a complete law of their distribution. We ask ourselves if such persisting difficulties could be understood as due to theoretical incompatibilities. We consider the problem in the conceptual framework of computational theory. This article is a contribution to the philosophy of mathematics proposing different possible understandings of the supposed theoretical unavailability and indemonstrability of the existence of a law of distribution of prime numbers. Tentatively, we conceptually consider demonstrability as computability, in our case the conceptual availability of an algorithm able to compute the general properties of the presumed primes’ distribution law without computing such distribution. The link between the conceptual availability of a distribution law of primes and decidability is given by considering how to decide if a number is prime without computing. The supposed distribution law should allow for any given prime knowing the next prime without factorial computing. Factorial properties of numbers, such as their property of primality, require their factorisation (or equivalent, e.g., the sieves), i.e., effective computing. However, we have factorisation techniques available, but there are no (non-quantum) known algorithms which can effectively factor arbitrary large integers. Then factorisation is undecidable. We consider the theoretical unavailability of a distribution law for factorial properties, as being prime, equivalent to its non-computability, undecidability. The availability and demonstrability of a hypothetical law of distribution of primes are inconsistent with its undecidability. The perspective is to transform this conjecture into a theorem.

2004

Let π(x) denote the number of primes smaller or equal to x. We compare √ π(x) with √ R(x) and √ ℓi(x), where R(x) and ℓi(x) are the Riemann function and the logarithmic integral, respectively. We show a regularity in the distribution of the natural numbers in terms of a phase related to ( √ π − √ R) and indicate how ℓi(x) can cross π(x) for the first time.