An Observation on Distribution of Prime Numbers
Dariush Bahrami2021
Sign up for access to the world's latest research
checkGet notified about relevant papers
checkSave papers to use in your research
checkJoin the discussion with peers
checkTrack your impact
Abstract
In this research first, a sequence of properties called delta is assigned to each prime number and then examined. Deltas are only dependent on the distribution of prime numbers, so the results obtained for the delta distribution can be considered as a proxy for the distribution of prime numbers. The first observation was that these properties are not unique and different prime numbers may have the same value of delta of a given order. It was found that a small number of deltas cover a large portion of prime numbers, so by recognizing repetitive deltas, the next prime numbers can be predicted with a certain probability, but the most important observation of this study is the normal distribution of deltas. This research has not tried to justify the obtained observations and instead of answering the questions, it seeks to ask the right question.
Related papers
On the statistical distribution of prime numbers: A view from where the distribution of prime numbers are not erratic
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. Keywords prime number generator, prime number theorem, twin primes, erratic nature of primes
Deciphering Prime Number Distribution: Insights from Natural Numbers
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
On the statistical distribution of prime numbers
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.
On the random nature of (prime) number distribution
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.
Statistical Distributions of Prime Number Gaps
Journal of advances in mathematics and computer science, 2024
A heuristic i.e. empirical approach to the problem of prime number gaps of many kinds and types, different degrees and orders, treated as simple raw experimental data from the statistical viewpoint is presented. The aim of the article is to show a picture of the actual situation of prime number gaps in order to describe and to try to understand the structure itself of prime gaps of various kinds and orders as well as of primes themselves. The data base comprises the finite sequences of prime number gaps up to the value Pn of the prime counter n = 5•10 7 that is P5E7 = P(5•10 7) = 982,451,653 all of them available in the net. The statistical distributions of prime gaps are best-fitted by the pseudo-Voigt fit function, a convolution of the Lorentz and the Gauss differential distribution functions, or by the so-called E-exp or exp-exp differential distribution function or by a log-linear histogram according to the kind of gaps examined, either δ i Pn (higher order gaps) or Δ k Pm = Pm-Pm-k (deltalags) with i and k ≥ 2 or the simple linear differences δ 1 Pm = Δ 1 Pm = ΔPm= Pm-Pm-1 respectively. One of the unexpected results of the investigation is the appearance of inner structures at high values of nΔ, the number of the intervals of the distributions, suggesting the presence of groups or clusters strictly linked to the nature of prime numbers themselves in which the same phenomenology is present.
The first digit frequencies of primes and Riemann zeta zeros tend to uniformity following a size-dependent generalized Benford's law
Arxiv preprint arXiv:0811.3302, 2008
Prime numbers seem to distribute among the natural numbers with no other law than that of chance, however its global distribution presents a quite remarkable smoothness. Such interplay between randomness and regularity has motivated scientists of all ages to search for local and global patterns in this distribution that eventually could shed light into the ultimate nature of primes. In this work we show that a generalization of the well known first-digit Benford's law, which addresses the rate of appearance of a given leading digit d in data sets, describes with astonishing precision the statistical distribution of leading digits in the prime numbers sequence. Moreover, a reciprocal version of this pattern also takes place in the sequence of the nontrivial Riemann zeta zeros. We prove that the prime number theorem is, in the last analysis, the responsible of these patterns. Some new relations concerning the prime numbers distribution are also deduced, including a new approximation to the counting function π(n). Furthermore, some relations concerning the statistical conformance to this generalized Benford's law are derived. Some applications are finally discussed.
On Prime Numbers and Related Applications
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.
0 30 92 51 v 1 1 0 Se p 20 03 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.
An Algorithmic Random Integer Generator based on Prime Numbers Distribution
2019
More than a philosophic thinking, we combine two researchers wishes on randomness reproduction and prime numbers distribution. Indeed, up to now we cannot rigorously answer the question on randomness of primes [6, page 1]. We then propose an example of algorithms that can be deduced by that connection. For this purpose, our main procedure uses prime gap sequence variation. An evaluation on randomness reproduction is made at the end for a conclusion about prime numbers distribution and its implications.
A Study on the Statistical Properties of the Prime Numbers Using the Classical and Superstatistical Random Matrix Theories
Advances in Mathematical Physics
The prime numbers have attracted mathematicians and other researchers to study their interesting qualitative properties as it opens the door to some interesting questions to be answered. In this paper, the Random Matrix Theory (RMT) within superstatistics and the method of the Nearest Neighbor Spacing Distribution (NNSD) are used to investigate the statistical proprieties of the spacings between adjacent prime numbers. We used the inverse χ 2 distribution and the Brody distribution for investigating the regular-chaos mixed systems. The distributions are made up of sequences of prime numbers from one hundred to three hundred and fifty million prime numbers. The prime numbers are treated as eigenvalues of a quantum physical system. We found that the system of prime numbers may be considered regular-chaos mixed system and it becomes more regular as the value of the prime numbers largely increases with periodic behavior at logarithmic scale.
Related topics
Related papers
submitter : A heuristic study of the distribution of primes in short and not-so-short intervals
2018
A numerical study on the distributions of primes in short intervals of length h over the natural numbers N is presented. Based on Cramér's model in Number Theory, we obtain a heuristic expression applicable when h log N but h N , providing support to the Montgomery and Soundararajan conjecture on the variance of the prime distribution at this scale.
A heuristic study of the distribution of primes in short and not-so-short intervals
arXiv: Number Theory, 2018
A numerical study on the distributions of primes in short intervals of length $h$ over the natural numbers $N$ is presented. Based on Cram\'er's model in Number Theory, we obtain a heuristic expression applicable when $h \gg \log{N}$ but $h \ll N$, providing support to the Montgomery and Soundararajan conjecture on the variance of the prime distribution at this scale.
Frequency Distribution of Prime Numbers between an Integer and its Square: A Case Study
Cornell University - arXiv, 2022
The chronicle of prime numbers travel back thousands of years in human history. Not only the traits of prime numbers have surprised people, but also all those endeavors made for ages to find a pattern in the appearance of prime numbers has been captivating them. Until recently, it was firmly believed that prime numbers do not maintain any pattern of occurrence among themselves. This statement is conferred not to be completely true. This paper is also an attempt to discover a pattern in the occurrence of prime numbers. This work intends to introduce some mathematical well-known equations that point to the existence of a simplistic pattern in the number of primes within the range of a number and its square. We assume that the rigorous evaluation of the perceived pattern may benefit in many aspects such as applications of encryption, algorithms concerning prime numbers, and many more.
Is there an Order in the Distribution of Prime Numbers
viXra, 2019
By arranging the prime numbers on four columns ten-to-ten (columns of one, three, seven, nine) and establishing a suitable correspondence between the quadruples obtained and the numbers between zero and fifteen, we obtain a synthetic representation of them which allows to establish that the order in the distribution of prime numbers among positive natural numbers is not random.
The Density of Primes Less or Equal to a Positive Integer up to 20,000: Fractal Approximation
Recoletos Multidisciplinary Research Journal, 2013
The highly irregular and rough fluctuations of the number of primes less or equal to a positive integer x for smaller values of x (x≤20,000) renders the approximations through the Prime Number Theorem quite unreliable. A fractal probability distribution more specifically, a multi-fractal fit to the density of primes less or equal to x for small values of x, is tried in this study. Results reveal that the multi-fractal fit to the density of primes in this situation outperforms the Prime Number Theorem approximation by almost 200% viz. the prediction error incurred by using the PNT approximation is double that of the multi-fractal fit to the density of primes. The study strongly suggests that a better multi-fractal distribution exists, even for large x, than the Prime Number approximation to the density of primes.
A study of divergence from randomness in the distribution of prime numbers within the arithmetic progressions 1+6n and 5+6n
arXiv: General Mathematics, 2018
In this article I present results from a statistical study of prime numbers that shows a behaviour that is not compatible with the thesis that they are distributed randomly. The analysis is based on studying two arithmetical progressions defined by the following polynomials: ($1+6n$, $5+6n$, $n\in{N}$) whose respective numerical sequences have the characteristic of containing all the prime numbers except $3$ and $2$. If prime numbers were distributed randomly, we would expect the two polynomials to generate the same number of primes. Instead, as the reported findings show, we note that the polynomial $5+6n$ tends to generate many more primes, and that this divergence grows progressively as more prime numbers are considered. A possible explanation for this phenomenon can be found by calculating the number of products that generate composite numbers which are expressible by the two polynomials. This analysis reveals that the number of products that generate composite numbers expressib...
Related Distribution of Prime Numbers
Studies in Mathematical Sciences, 2013
The surplus model is established, and the distribution of prime numbers are solved by using the surplus model.
Information entropy and correlations in prime numbers
2003
The difference between two consecutive prime numbers is called the distance between the primes.
On Stochastic Approximations to the Distribution of Primes and Prime Metrics
Journal of Higher Education Research, 2017
MATHEMATICS OF HUNG-PING TSAO III: THREE MARVELOUS DISTRIBUTION MODELS OF SMALL PRIME NUMBERS AS TERMINAL VALUES IN THE PROCESS OF SUCCESSIVELY SUMMING UP ALL FACTORS OF THE NATURAL NUMBERS
"Evolutionary Progress in Science, Technology, Engineering, Arts, and Mathematics (STEAM)", 2024
Hung-ping Tsao (2024). Mathematics of Hung-ping Tsao III: Three Marvelous Distribution Models of Small Prime Numbers as Terminal Values in the Process of Successively Summing up All Factors of the Natural Numbers. In: "Evolutionary Progress in Science, Technology, Engineering, Arts, and Mathematics (STEAM)", L. K. Wang and H.P. Tsao (eds). Vol. 6, No. 1, Jan. 2024; 100 pages. Lenox Institute Press MA, USA. ..... ABSTRACT: We present here three distribution models M1 (for n up to 1000), M2 (for n up to 10000) and M3 (for 2n up to 10000) of the small prime numbers as terminal values in the new platform (see [14]), where the terminal value t(n) for each natural number n being defined by way of successively summing up the prime factors. By defining A = {5} and B = {7, 11, 13}, we call 5 the A-terminal value, 7, 11 or 13 a B-terminal value and all others C-terminal values, where C = {1, 2, 3, 4, 17,19, 23, 29, …}. Our findings are summarized as follows. For M1, the frequency for each small prime number to occur as terminal values remains almost the same as n increases from 1 to 1000, namely 30% for 5, 17% for 7, 7% for 11 and 7% for 13. Moreover, the frequencies of singles, doubles, triples and quadruples respectively from A and B are almost identical and they combine to 61% for singles, 16% for doubles, 4% for triples and 1% for quadruples. For M2, the frequency for each small prime number to occur as terminal values remains almost the same as n increases from 1 to 10000, namely 28% for 5, 16% for 7, 6% for 11 and 7% for 13. Moreover, the frequencies of singles, doubles, triples and quadruples respectively from A and B are almost identical and they combine to 57% for singles, 16% for doubles, 4 % for triples and 1% for quadruples. We also come up with a good way of estimating the frequency of singles from A and B combined by using twin primes (tp, tp+2) which will be called cheese sandwich if the terminal value of tp+1 is from C or ham sandwich otherwise.
Loading Preview
Sorry, preview is currently unavailable. You can download the paper by clicking the button above.