Exploring Anomalous Relaxation Models in Prime Number Distribution and Their Relevance to Sustainable Development Education

Exploring Anomalous Relaxation Models in Prime Number Distribution and Their Relevance to Sustainable Development Education

Sara AlauLdeen Abdulqader Israa Ibraheem Al_Barazanchi Zahraa A. Jaaz Ravi Sekhar* Pritesh Shah Shilpa Malge

Department of Registration and Students Affairs, University of Technology, Baghdad 10001, Iraq

Department of Communication Technology Engineering, College of Information Technology, Imam Ja'afar Al-Sadiq University, Baghdad 10001, Iraq

College of Computing and Informatics, Universiti Tenaga Nasional (UNITEN), Kajang 43000, Malaysia

Computer Department, AlNahrain University, Jadriya, Baghdad 10001, Iraq

Symbiosis Institute of Technology (SIT) Pune Campus, Symbiosis International (Deemed University) (SIU), Pune 412115, India

Corresponding Author Email: 
28 December 2023
17 March 2024
25 March 2024
Available online: 
30 May 2024
| Citation

© 2024 The authors. This article is published by IIETA and is licensed under the CC BY 4.0 license (http://creativecommons.org/licenses/by/4.0/).



Mathematical frameworks termed anomalous relaxation models are used to characterize the distribution of prime numbers in an unconventional way. Many scholars need clarification on the challenges with prime numbers (PN), because of their mysterious and complicated qualities because it is challenging to build a model that characterizes the relaxed distribution of PN in a collection of all natural numbers. Although certain qualitative methods, like Riemann's Theory, exist to the best of the educational knowledge can characterize the distribution of the prime numbers, the law controlling the distribution of the PN is unclear, and the correct model needs to be established. The distribution of prime numbers is obtained in the current work for the first time using the anomalous relaxation model with the fractional derivative. The new model matches well with the data of PN, according to comparisons with Riemann's Theory and the PN Theory of sustainable development education.


prime number, Riemann’s Theory, relaxation model, natural number, fractional derivatives, sustainable development education, environmental education

1. Introduction

This think about clarifies that prime numbers have puzzling and complicated qualities, which makes it troublesome to construct a show that characterizes their dissemination. Prime numbers are special in that they can as it were be isolated by 1 and themselves, and they don't take after any unsurprising design. This makes it challenging to set up a numerical show that precisely characterizes their dissemination in a collection of all characteristic numbers. Whereas certain subjective strategies like Riemann's Hypothesis exist to characterize the dispersion of prime numbers, the law controlling their distribution is still vague, and the right demonstrate must be set up. This can be where bizarre unwinding models come in, as they offer a better approach to characterize the conveyance of prime numbers in an unusual way. Mathematicians have been fascinated by the distribution of prime numbers, an essential topic in number theory. Prime numbers are integers greater than one that can be divided by themselves and one. Even though there are many prime numbers among the natural numbers, their distribution is extremely unpredictable and puzzling. Mathematicians have been unable to comprehend the patterns determining the distribution of prime numbers within the diverse range of natural numbers [1]. Through a variety of methods, mathematicians have attempted to understand the distribution of prime numbers, including the revolutionary work of Bernhard Riemann and well-known Riemann Hypothesis in the 19th century. Through developing the Riemann zeta function, which is related to the behaviour of prime numbers, Riemann's Theory gave significant insight into the distribution of primes [2]. The dispersion of prime numbers raises a number of concerns in spite of incredible advance. Concurring to the Twin PN Guess, there are numerous prime pairings with an indistinguishable two-prime contrast is the foremost well-known unsolved issue in this field [3]. A thorough exhibit is unattainable considering the reality that a few machine endeavors have given information to support this guess. Within the space of prime numbers, a scientific approach that employments bizarre unwinding components to clarify and comprehend the conveyance of prime numbers is alluded as an atypical unwinding demonstrate [4]. These numerical material science and number hypothesis models, which have been created, and outlined to shed modern light on the conduct of prime numbers, which had been examined utilizing the conventional strategies like Riemann zeta work and probabilistic models [5]. Distinctive fragmentary differential conditions and related numerical strategies can be utilized in unusual unwinding models to characterize the prime number dispersion. They endeavor to clarify the madnesses, varieties, and clustering that can be seen within the dissemination of prime numbers, but that will not be tended to by conventional models [6].

Twin primes, agreeing to the hypothesis of prime numbers are sets of prime numbers that vary from one another by absolutely two Twin primes known as these pairings are a captivating concept in number hypothesis. The Twin Prime Guess asserts that are inconclusively numerous twin prime sets, in spite of the fact that this can be an open numerical address [7]. Mathematicians have been fascinated by twin primes for generations, and their investigation is a key component of research into the mathematics of prime numbers and number theory with connections to hard concepts like the distribution of primes and screening methods [8]. The exploration of anomalous relaxation models in prime number distribution aims at the development of innovative mathematical structures that can provide more precise and complex perspective on the distribution of prime numbers among natural numbers, possibly revealing hidden patterns and irregularities in this fundamental area of number theory.

2. Related Works

Berdondini [9] suggested that the method uses the Z-relative number set to evaluate the distribution of prime numbers. When using negative numbers as well, it becomes clear that the configuration reduces the gap between the two powers with the same actual base D value, with |D|≤P was the organization of 2P+1 sequential numbers from -P to P. Their arrangement was significant because it was anticipated to reduce redundancy between the prime numbers under consideration by enhancing the total number of prime numbers in a collection of successive integers.

Liu [10] discussed the distribution of twin prime numbers. It turns out that the related prime distribution rate of a composite number achieves the respective prime distribution rate of integers by combining the corresponding prime distribution rates of integers and composite numbers. It includes a discussion of the distribution of prime numbers that correlate to integers and composite numbers. Thus, the total number of twin prime numbers was determined. It offered a useful method for researching the theory of twin prime numbers.

Gensel [11] provided a fresh approach to demonstrating the twin prime conjecture, remove all twin primes that exist from the current level of the twin primal generators. Due to each pair's identical generator, twin contributions can be identified from each other. Attempting to identify the Twin Primes at the same level as their creators makes sense as a consequence. They specify the "pn-numbers"x was a set of numbers, and those numbers were prime. 6x+1 and 6x-1 were coprime. They can indirectly demonstrate the Twin Prime Hypothesis by considering a typical distance of (pn) between the (pn) numbers.

Carbó-Dorca [12] created a straightforward formula to get primes from the constituents of the prime natural number set. The current method uses a recursive algorithm to add powers of 2 to earlier subsets in order to produce natural numbers. According to unprocessed data, one can create a prime number series in their manner starting from an odd natural integer. One can conjecture that any prime number can be produced by adding a prior prime number to an element or by adding the items in the second set.

Nandanwar [13] identified a precise range of number series where twin primes were not random. They were part of an infinitely repeating cycle in those kinds of ranges that range was a part of cycle in their series' recurring pattern. They succeeded in demonstrating that there was a limited twin prime between the squares of any two consecutive odd numbers as well as any two consecutive prime numbers, which can be demonstrated because the following odd number will be p+2 in the lower bound equation. Their method can be used to construct lower bounds for conjectures of a similar nature. Fiori [14] created formulas and techniques that maximize the transformation of implicit as well as explicit derived from the PN algorithm. Their approaches were strong and innovative because they took advantage of Mathematical findings with a small x to improve both the established quantitative bounds and the basic predetermined estimations. The framework's PN theorem and other traditional forms exceed those approaches' various variables and lengths. Maïsseu [15] suggested that two coefficients, j, and i related to their subscript NP, which was determined by the equation $\mathrm{P}=6 N_P \pm 1$, used to break down natural numbers, P. NP was a natural whole number. There were two times six subsets of the prime numbers or composite numbers that were described by the equation $\mathrm{P}=6 N_P \pm 1$. It was believed that natural whole numbers cannot be broken. The term "composite numbers" refers to those that can be divided, whereas "breaking" natural whole numbers was not possible.

De [16] provided an elementary proof of the well-known Prime Number Theorem, they would examine the outstanding contributions of Norwegian mathematician Selberg and Hungarian mathematician Erdos. They would focus on investigating and finding appropriate estimations for the Chebyshev Theta Function $\vartheta(x)$, in addition to exposing themselves to the concept of arithmetic functions. Additionally, they'll attempt to determine easy-to-implement qualities of another function $\rho(x)$.

Zaman [17] created a link between traditional and contemporary prime numbers using upper and lower bounds. They also present an innovative approach for computing the sum of prime integers. In-depth outcomes, such as the prime number theorem's insights into prime gaps, emerged from the search for understanding the distribution of primes. They bridge the gap between theorems and contemporary advancement by integrating these upper and lower bounds.

Machado and Lopes [18] utilized mathematical and information visualization tools to investigate prime numbers (PN) from the viewpoint of complex systems (CS). The Canberra, Euclidean, Jaccard, and Lorentzian distances were the four unique metrics used to evaluate the differences between these objects. The information was processed using the multidimensional scaling Multidimensional scaling (MDS) technique. The representations provide an innovative method for employing modern scientific visualization to investigate the complex subject of PN.

Korneev et al. [19] identified an accurate formula for calculating the number of composites and, the number of primes with a set of eight arithmetic progressions (APs), determining that the APs together contain all prime numbers other than 2, 3, and 5 each AP's composites were calculated from the corresponding AP's primes. The set that includes positive numbers can be subdivided into a variety of sets, including sets of even numbers, odd numbers, primes, and composite numbers. All three formulas estimate the general population of positive integers' number of primes and one of them generates an extremely accurate upper-bound estimate.

Aiazzi et al. [20] identified the twin primes contained inside the prime gaps were examined. The existence of primes would support the hypothesis that they cannot end because the bounds of these gaps were set by matches of prime numbers, allowing for an approximation of the twin primes up to infinity. By using a similar approach to the exponential integral function Li(x), their formulation may calculate the prime number function. By considering the arrangements with an estimate of two, the current examination was extended to incorporate the twin primes.

3. Methodology

The examination of prime numbers has captivated mathematicians for centuries, as the strange nature of these integrability proceeds to astound specialists. Conventional strategies for examining prime number conveyance have centered on deterministic designs and normality, but atypical unwinding models offer an interesting approach that highlights the haphazardness and abnormality characteristic in prime number groupings. Odd unwinding models are established in measurable material science and the hypothesis of dynamical frameworks, and they give a system for examining the behavior of prime numbers in connection to their neighboring integrability. By consolidating concepts from factual material science, such as vitality flow and unwinding time, analysts have shed modern light on the dispersion of primes inside an arrangement of integrability. These models permit analysts to examine the factual properties of prime number groupings, uncovering designs that were already covered up.

The dispersion of prime numbers could be a complex and captivating point, and analysts proceed to investigate unused strategies for understanding this marvel. A few of the systems and approaches that have been utilized to think about prime number conveyance incorporate atypical unwinding models, upper and lower bounds, and number-crunching movements. These systems include a combination of concepts from factual material science, dynamical frameworks theory, and number hypothesis, and they point to supply unused bits of knowledge into the conveyance of prime numbers and its fundamental designs.

According to an early idea known as the Prime Number concept (PN), the probabilities of prime numbers are based on the logarithmic connection that exists between the total expected number and the prime numbers. Legendary hypothesized that the total amount of prime numbers not taken above, which increases the PN’s denominator's variable count.

$M$, is $\pi(M)=\frac{M}{B \ln M+A}$              (1)

Subsequently, changed it into:

$\pi(M)=\frac{M}{\ln M+B(M)^{\prime}}$               (2)

Prime numbers multiplied by integral form, in terms of their total,

$\lim _{m \rightarrow \infty} B(M)=-1.08366 \ldots$

Eleven years later, the expansion of Gauss's equation for the total number of prime numbers was stated using the zeta function $\varsigma(\mathrm{w})$.

$\pi(M) \sim L i(M)=\int_0^M \frac{d t}{\ln s}$.             (3)

It is obvious that there is no differential equation model and that the Eqs. (1)-(3) are descriptive formulations.

$Q(M)=\sum_{m=1}^{\infty} \frac{\mu(m)}{m} L i\left(M^{\frac{1}{m}}\right)$               (4)

In order to address mathematical problems, physics and mathematics are used more and more. We consider the geographic distribution of the highest possible number as a method of physical relaxation.

The total typical number increases, the proportion of prime numbers falls. It behaves in a long-tailed manner and rejects the exponential scaling law. We seek to utilize a fractional breakdown model to describe the PN geographical distribution because fractional relaxing models can explain some energy breakdown behaviour in complicated materials.

3.1 Concepts relating to prime number distribution

The following definitions apply to the number of prime numbers that don't exceed N:

$\pi(M)=\sum_{j=1}^M O(j), O(j)=\left\{\begin{array}{cc}1, & j \text { is a prime number } \\ 0, & \text { else }\end{array}\right.$             (5)

Figure 1 illustrates that PN density has changed from $v(M)=\pi(M) / M$. It is evident that the logarithmic value of the overall natural numbers increases, the proportion of PN falls. The exponential law is broken by the long-tailed behavior of this concentration curve.

PN concept, an early explanation of the geographic distribution of prime numbers, claims that:

$\pi(M) \sim \frac{M}{\ln M}$                (6)

Eq. (6) has the characteristic that, when M is getting close to $+\infty$:

$\lim _{m \rightarrow \infty} \frac{\pi(M)}{\frac{M}{1 m(M)}}=1 \frac{1}{\ln } M$                    (7)

The differential expression regulates the PN's concentration of prime numbers, which is $1 / \ln M$.

$\left\{\begin{array}{c}\frac{d u(t)}{d s}+B v^2(t)=0, t>0 \\ v(0)=1\end{array}\right.$             (8)

In this model, the initial value of 1 has statistical significance and does not imply because the number that begins with 1 is a PN. The prime probability density is taken into account using the power law $v=B t^{-1}$ in this method. In Figure 2, it is smaller than the entire natural number 1021, yet the standard variance range's value in absolute terms is between 100 and 10-2.  

Figure 1. (a) PN density; (b) The PN density in logarithmic scale

Figure 2. (a) PN theory results; (b) Absolute relative inaccuracy according to the prime number theory

Figure 3. (a) Riemann’s result; (b) The Riemann theory's relative inaccuracy in terms of absolute value

The infinite sequence form of the prime number formula yielded more precise result,

$Q(M)=1+\sum_{l=1}^{\infty} \frac{(\ln M)^l}{l!l \zeta(l+1)}$                   (9)

$\zeta(t)=\sum_{m=1}^{\infty} \frac{1}{m^t}=\underset{o}{ }\left(1-\frac{1}{o^t}\right)^{-1} Q(t)>1$               (10)

The proportionate condition for Condition (3) is the formula. The interaction of R(M) and the genuine information is appeared in Figure 3(a), and the whole sum of the relative mistake is appeared in Figure 3(b). Concurring to Figure 3(b), the relative inaccuracy drops as the whole number of normal numbers rises. Be that as it may, due to this hypothesis containing the vital differential condition show and basically factual approach with interminable arrangement shape, it is incapable to illustrate the characteristics of the prime conveyances and the PN fundamental numerical run the show.

3.2 Model for prime numbers distribution

They appear to follow the law of chance as they spread like weeds among natural numbers, and it's impossible to anticipate the next one will emerge. The distribution of PN is described in an excellent graphic way. As far as the researchers' knowledge goes, the law controlling the distribution of PN remains unclear, and the equation that governs the model has not been established. Some physical occurrences in the natural world display traits that are similar to those of the prime distribution. In the meanwhile, Figure 1 shows that the emission mechanism constitutes changing the PN density. As a result, using a physical model can be more effective to explain the distribution of PN and disclose its amazing regularity. As a result, we want to use the anomalous relaxation equation model in this inquiry to depict the distribution of prime numbers in an unexpected manner.

3.2.1 Relaxation model

One such system is the bizarre unwinding show, which has been utilized to consider prime number conveyance. Bizarre unwinding models are established in measurable material science and the hypothesis of dynamical frameworks, and they offer an interesting approach to understanding the conveyance of prime numbers.

Not at all like conventional strategies, which center on deterministic designs and consistency, odd unwinding models highlight the haphazardness and abnormality inborn in prime number groupings. By joining concepts from measurable material science, such as vitality elements and unwinding time, analysts have shed modern light on the dissemination of primes inside a grouping of integrability. Bizarre unwinding models give a system for examining the behavior of prime numbers in connection to their neighboring integrability, and they permit analysts to examine the measurable properties of prime number arrangements, uncovering designs that were already covered up. These models have driven to exceptional disclosures and experiences into the conveyance of prime numbers, and they proceed to be a dynamic zone of investigate in arithmetic and number hypothesis. This ponder of prime number dispersion may be a complex and interesting subject, and analysts proceed to investigate modern strategies for understanding this marvel. By investigating odd unwinding models and other numerical systems, analysts trust to pick up a more profound understanding of the dissemination of prime numbers and its suggestions for other zones of science and innovation.

In this examination, we see at the PN conveyance as a physical unwinding prepare that takes after the push unwinding for deformable persistent substance pressure conditions. The basic rule of one of the conventional models for materials having elastic characteristics that can clarify this relaxing feature as follows:

$\sigma+\tau \frac{d \sigma}{d t}=\eta \frac{d \varepsilon}{d t}$              (11)

where, $\tau=\frac{\eta}{H_0}$ equilibrium compliance, $(\sigma$ signifies stress, $\varepsilon$ denotes strain, $\eta$ viscosity coefficient). For ongoing tension, we can achieve the stress relaxation $\left(\varepsilon=\varepsilon_0=\right. const )$,

$\sigma(s)=\sigma_0 f^{-\frac{s}{\tau}}$                (12)

This complies with the differential equation.

$\frac{d}{d t} f(s)+\tau^{-1} f(s)=0$                   (13)

By using the Fourier transform, one may determine the response time function.

$\tilde{f}(j \omega)=\frac{1}{\frac{1}{\tau}+j \omega}$              (14)

However, Maxwell's relaxation law was unable to suit the experiment's data. Significant improvements were made to the development or generation of the decay rules, to explain the relaxation processes. Relaxing under pressure fits into a power law $\sigma(s)=\sigma_0 e^{-\left(\frac{s}{\tau_0}\right)^\beta}$, a large number of mediums and dielectrics' distribution and absorption are described.

$\tilde{f}_\alpha(j \omega)=\frac{1}{b_0+a(j \omega)^\alpha}, \quad 0<\alpha<1$                 (15)

The partial derivative equations are promoted as a practical method for describing the movement of complex systems that are subject to unusual diffusion processes and non-exponential relaxing patterns. Following a description of the anomalous relaxing model with partial time derivative:

$D_s^\alpha f(s)+B f(s)=0, \quad 0<\alpha<1$                (16)

where, the stress or energy is represented by f(s), B stands for the relaxation coefficient, and $C_s^\alpha$. The fractional derivative of Caputo is defined as:

$\begin{aligned} & C_s^\alpha v(s)= \frac{1}{\Gamma(m-\alpha)} \int_0^s \frac{t^{(n)}(\tau) d \tau}{(t-\tau)^{\alpha+1-m}}, (m-1<\alpha<m)\end{aligned}$              (17)

where, α is the fractional derivative's order. Eq. (17) switches to the Maxwell model's typical relaxing when α=1. The proportional derivative's Fourier transform is described as:

$F\left[{ }_{-\infty}^D C_s^\alpha e(s) ; \omega\right]=(j \omega)^\alpha \tilde{e}(\omega)$              (18)

Implementing the Fourier transform to the variable s in Eq. (16),

$E_b(w)=\sum_{l=0}^{\infty} \frac{w^l}{\Gamma(\alpha l+1)}$                  (19)

3.2.2 Prime density model

We suggest that the controlling equation of motion for the fundamental density be Eq. (17).

$\left\{\begin{array}{c}C_q^\alpha v(q)+B v(q)=0 \\ v(0)=1\end{array}\right.$              (20)

where, $q=[\log (\mathrm{M})]^{1 / \alpha}$, M stands for the quantity of a natural number. In this model, the initial value has statistical relevance. The Mittag-Leffler function is the response to Eq. (20).

$E_\alpha(-B \log (M))=\sum_{l=0}^{\infty} \frac{[-B \log (M)]^l}{\Gamma(\alpha l+1)}$              (21)

Figure 4. (a) Mittag-Lefller function result; (b) Absolute value of relative error

Figure 5. The absolute value of relative error in three methods

In Figure 4(a), the Mittag-Leffler function curve is shown and in Figure 4(b), the absolute magnitude of the relative inaccuracy between the M-L functions and the average density of the PN is shown. Figure 4(b) demonstrates that function $E_\alpha(-B \log (M))$ can give an adequate account of the quantity of prime numbers. Comparative error's absolute magnitude is acceptable and declines the natural number rises.

A contrast of the three approaches is shown in Figure 5. The distribution of values of the prime is described by Riemann's Method, but there is no equivalent to the differential equation model. The (PN) provides the differential equation and explains the prime probability density changes, according to a power law, it has the drawback of having an excessively large inaccuracy. The function of probability density obtained using the partial relaxation equation model is more accurate and can represent the pattern of the probability density decrease for prime numbers.

The examination of prime numbers has captivated mathematicians for centuries, as the secretive nature of these integrability proceeds to confuse specialists. In later a long time, analysts have dug into the world of bizarre unwinding models to pick up a more profound understanding of prime number dispersion. These models, established in measurable material science and the hypothesis of dynamical frameworks, offer a one of a kind approach to unraveling the secrets encompassing prime numbers.

Bizarre unwinding models give a system for considering the behavior of prime numbers almost their neighboring integrability. Not at all like conventional strategies, which center on deterministic designs and consistency, these models highlight the arbitrariness and abnormality inalienable in prime number arrangements. By uniting concepts from quantifiable fabric science, such as essentialness components and loosening up time, examiners have shed unused light on the scattering of primes interior a course of action of integrable.

The examination of odd loosening up models has driven to extraordinary revelations and bits of information into the spread of prime numbers. These models allow examiners to look at the genuine properties of prime number courses of action, revealing plans that were as of now secured up. The combination of dynamical systems speculation and real fabric science has illustrated to be a profitable approach for understanding the complexities of prime number scattering. The consider of atypical loosening up models in prime number scattering offers a promising way for unraveling the astounding properties of these integrable. The integration of concepts from quantifiable fabric science and dynamical systems speculation gives an unused point of see that uncovers secured up plans and clarifies the complicated nature of prime number groupings. With proceeded inquire about and investigation, the bits of knowledge picked up from these models may clear the way for advance progressions in number hypothesis and contribute to a more profound understanding of prime numbers. The article talks about the dissemination of prime numbers, a basic theme in number hypothesis. The conveyance of prime numbers is eccentric and astounding, and mathematicians have been incapable to get it the designs that decide their dissemination inside the extend of common numbers. The article investigates bizarre unwinding models, a numerical approach that employments abnormal unwinding components to clarify the dissemination of prime numbers. These models point to clarify the madnesses, varieties, and clustering watched within the dissemination of prime numbers, which conventional models may not address.

The ponder presents an unused demonstrate that characterizes the dissemination of prime numbers in an unusual way utilizing atypical unwinding models with fragmentary subordinates. This unused show adjusts well with the information of prime numbers, as compared to Riemann's Hypothesis and the prime number hypothesis. It offers a new point of view on the conveyance of prime numbers among characteristic numbers, possibly uncovering covered up designs and inconsistencies in this principal range of number hypothesis. The ponder moreover talks about the Twin Prime Guess, the foremost well-known unsolved issue within the field of prime numbers. This guess states that there are boundlessly numerous twin prime sets, in spite of the fact that it remains an open address in arithmetic. By investigating odd unwinding models in prime number dissemination, the think about points to create inventive scientific structures that can give a more exact and comprehensive understanding of the dissemination of prime numbers among characteristic numbers. This examination may uncover covered-up plans and anomalies in this fundamental run of number theory. In common, this think approximately presents an inquisitively and illuminating perspective on the transport of prime numbers, a noteworthy subject in number theory. The examination of strange loosening up models in prime number scattering offers a present day perspective on this foremost extend of think almost and has the potential to lead to future encounters and divulgences.

4. Conclusion

Odd loosening up models offer a uncommon approach to analyzing prime number spread that contrasts from other logical frameworks. Ordinary methodologies for analyzing prime number scattering habitually center on deterministic plans and typicality, seeking out for to recognize numerical conditions or conditions that can absolutely expect the scattering of prime numbers. In any case, prime numbers are broadly troublesome to anticipate, and their scattering appears up to be significantly subjective and intermittent. Odd loosening up models take a different approach by emphasizing the assertion and anomaly characteristic in prime number groupings. By joining concepts from truthful fabric science, such as essentialness components and loosening up time, these models provide a framework for analyzing the behavior of prime numbers in association to their neighboring integrable. This permits analysts to examine the factual properties of prime number arrangements and uncover designs that were already covered up. In general, bizarre unwinding models offer a better approach to approach the ponder of prime number dispersion that takes under consideration the interesting qualities of prime numbers and their dispersion.

The vitality scattering and unwinding in complex materials are regularly depicted utilizing the atypical unwinding demonstrate with fractional subsidiaries. The dissemination of prime numbers, by coincidence incorporates a comparative long-tailed slant and rejects the exponential law. As a result, we consider the prime number dispersion behavior as a bizarre unwinding process. The results of the numerical reenactment illustrate the atypical unwinding show can capture the advancement of the prime thickness. The prime thickness depends on and takes after the Mittag-Leffler rot as the whole number of common numbers increases $E_\alpha(-B \log (M))$. This comes about from the determination of the fragmentary subordinate of the bizarre unwinding condition. Anomalic unwinding models in prime number dispersion are troublesome to work, due to their complexity and need of empirical validation, and their utility is compelled without considerable prove and hypothetical approval due to the essential nature of the issue. However, it is worth noting that this is an active area of research in mathematics and number theory, and there are likely to be many ongoing efforts to better understand the distribution of prime numbers and its implications for other fields. Some potential areas of future work might include developing new mathematical models or frameworks for studying prime number distribution, exploring the statistical properties of prime number sequences, or investigating the connections between prime numbers and other areas of mathematics or science. Additionally, there may be efforts to apply knowledge of prime number distribution to practical problems in fields such as cryptography or computer science. Overall, the study of prime number distribution is a rich and complex area of research with many potential avenues for future exploration and discovery.


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