Prime Number Theory Research Articles

New prime number theory

This paper introduces a novel approach to estimating the sum of prime numbers by leveraging insights from partition theory, prime number gaps, and the angles of triangles. The methodology is applied to infinite sums and the nth sum, and several ways of defining the nth sum of a prime number are proposed. By using the Ramanujan infinite series of natural numbers, it is possible to derive an infinite series of prime numbers.

Investigations on Application of Probabilistic and Mathematical Computing in Design and Statistical Analysis of Lightweight Cryptography

This research provides in-depth investigation into the application and role of probability and mathematics in lightweight cryptography featuring IoT software. The IoT device fleet growth is continuously accompanied by its resources, often poor. It creates a critical need for cryptographic solutions that will be effective and a heavy burden. The research article underlines the use of latest probabilistic models and mathematical concepts to create cryptographic algorithms that are cryptographically powerful and IoT-embedded devices can process them. The study commences with its deep mathematics investigation and focuses on consequently essential permutations, combinations, modular arithmetic and prime number theory, all of which represents the cornerstone of strong cryptographic algorithms development. The article then proceeds to demonstrate the use of probabilistic tools for scrutinizing the security and the effectiveness of these algorithms. A novel approach to grasping the vulnerabilities that may occur in the course of cryptographic implementations is also brought to light. A remarkable part of the investigation is toward designing small-sized cryptographic algorithms by using methods of mathematics computations to decrease the size of key, complexity and implementations of increased safety margin. The research process involves a thorough statistical demonstration of the algorithmic performance in several agendas of IoT exposing the resistance of the algorithms against cryptographic issues and the intensity of their resource consumption on ordinary IoT devices. By conducting these investigations, the paper contributes a lot of useful information on the harmonious relationship between probability methods, mathematical computing, and cryptographic development as well as providing the solid foundation for creating a lightweight cryptography which can be used in the Internet of Things (IoT) and other similar environments that have their unique needs and constraints. However, it not only advances the field of cryptography itself, but also outlines a practical roadmap on how to design secure, efficient cryptosystems, that can be used in its subsequent applications in the small but fast growing IoT technology landscape.

On the distribution of αp^2 + β modulo one for primes p such that p + 2 has no more two prime divisors

A classical problem in analytic number theory is to study the distribution of frac- tional part αp^k + β, k ≥ 1 modulo 1, where α is irrational and p runs over the set of primes. We consider the subsequence generated by the primes p such that p + 2 is an almost-prime (the existence of infinitely many such p is another topical result in prime number theory) and prove that its distribution has a similar property.

The development of P-adic numbers theoretically and their use in number theory

Stemming from the need to formalize the divisibility of rational numbers, p-adics provide a unique mathematical perspective rooted in prime number theory. This article provides a comprehensive discussion of p-adic numbers, including their research background, definition, properties, theory, and extensions. This study first elucidates the historical background and significance of p-adic numbers, emphasizing their key role in number theory and its applications. At the heart of this research is a deep dive into the precise definition of p-radical numbers, revealing their unique, often counterintuitive, distance measure. We look at their fundamental characteristics, demonstrate their illogical characteristics, and discuss how they affect transcendental number theory, Diophantine equations, and algebraic number theory. Furthermore, this article explores the theoretical foundations of p-radical numbers and their extensions, emphasizing their integral role in advanced mathematical structures such as p-radical analysis and p-radical geometry. By covering these aspects, this study aims to highlight the lasting impact of p-adic on modern mathematics, reshape our understanding of divisibility, and advance mathematical inquiry into new and uncharted territory.

Searching for nontrivial zeros in finite field carrying geometric metric ratios

With much emphasis on how to solve one of the toughest mathematical problems in field of prime number theory. There is a need to find a good coherent solution that gives much understanding into how to better unravel and solve the Riemann Hypothesis in the best way.

“Criticality” in the counting function of prime numbers: Theory and application to nuclear magic numbers

In this paper, considering critical phenomena and their phase transitions within the frame of the set of prime numbers, is attempted. Thus, the novel, theoretical and purely mathematical Model of Criticality based on Prime Numbers is introduced. This approach allows for the emergence of a parallelism between the physical concept of criticality and the corresponding concept in prime number theory. Based on this parallelism an application of the proposed model in determining the known magic numbers of Nuclear Physics is presented. This application introduces a physical meaning to the exceptional in properties set of the prime numbers and the corresponding prime number theory. Finally, going beyond the proposed model and its application, we suggest investigating other prime numbers as doubly magic numbers or candidate magic numbers for further experimental research.

Unifying colors by primes

RGB and CYMK are two major coloring schemes currently available for light colors and pigment colors, respectively. Both systems use letter-based color codes that require a large range of values to represent different colors. The problem is that these two systems are hard to use for manipulating any operations involving combinations of colors, and they lack the capacity for inter-changeability or unification. Based on prime number theory and Goldbach’s conjecture, this study presents a universal color system (C235) using a number-based structure to encode, compute and unify all colors on a color wheel. The proposed C235 system offers a unified representation for the efficient encoding and effective manipulation of color. It can be applied to designing a high-rate LCD system and colorizing objects with multiple attributes and DNA codons, opening the door to manipulating colors and lights for even broader applications.

CONTRIBUTION OF P.L. CHEBYSHEV TO THE DEVELOPMENT OF PRACTICAL MECHANICS IN RUSSIA

The scientific creativity of academician P.L. Chebyshev (1821-1894) is connected with the history of the development of practical mechanics, which is the application of mathematical principles to the construction of machines. In addition to outstanding works on the theory of prime numbers, probability theory and other areas of mathematics, P.L. Chebyshev founded the mathematical theory of machine synthesis and developed a number of important concepts of mechanisms. (Research purpose) The research purpose is presenting the program of P.L. Chebyshev's scientific activity aimed at the search for a mathematical theory of machine synthesis; to conduct a historical and scientific study of the problem of the genesis of P.L. Chebyshev's interest in applied mechanics. (Materials and methods) Studied the original works of P.L. Chebyshev, materials of historical and scientific research in domestic and foreign literature, archival information. (Results and discussion) The questions of the genesis of P.L. Chebyshev's interest in the development of practical mechanics were investigated. Among the factors that influenced the active activity of P.L. Chebyshev and outstanding results in the field of practical mechanics, the following were considered: interaction with N.V. Brashman during his studies at Moscow University; study of classical works by J.-V. Poncelet; personal contacts during foreign business trips and further interaction with famous European mathematicians and mechanics of that time; visiting and detailed study of the museums of machines and models in Paris, Brussels, London. We paid attention to the goal that the scientist set for himself - to minimize costs and achieve the greatest efficiency in the operation of machines. This is confirmed by the preserved sample of a continuous mechanical calculating machine, developed and created with the direct participation of P.L. Chebyshev. (Conclusions) P.L. Chebyshev made an outstanding contribution to the development of practical mechanics in Russia, laid the foundations of the theory of synthesis of mechanisms, became the founder of the theory of the structure of plane mechanisms.

A New Symbolic Time Series Analysis Method Based on Time-to-Space Mapping, through a Symmetric Magnetic Field, Quantized by Prime Numbers

This work presents a new analysis method for two-symbol symbolic time series based on the time-to-space mapping achieved through a device of current carrying circular rings. An algorithm based on the theory of prime numbers is proposed for the approximate estimation of the stratified magnetic field produced by the aforementioned device. The main property of the specific algorithm is that it quantizes the stratified magnetic field. If a two-symbol symbolic time series is used to determine the flow directions of the rings’ currents, a time-to-space mapping of the dynamics of the system producing the time series is observed. A unique “fingerprint” of the symbolic dynamics is shaped by the spatial allocation of the values of the six-valued symmetric quantized magnetic field produced by the device. This allows for the quantitative evaluation of the original system’s dynamics by analyzing the resultant quantized magnetic field values space allocation, in a spectrum ranging from the lack of dynamics (randomness) to the presence of dynamics at all scales (criticality). Two examples of application–corresponding to the extremes of the dynamics spectrum, specifically, for symbolic time series resulting from (a) a random numbers generator and (b) the spin alternation of 2D-Ising in its critical state, verify the reliable time-to-space mapping of the involved symbolic dynamics. Moreover, an application to the symbolic sequence produced by the DNA of the GAPDH (Glyceraldehyde-3-Phosphate Dehydrogenase) human gene is presented as a real-world, intermediate dynamics case. The proposed symbolic time series analysis method presents the advantage that can take into account information related to both symbols, which is particularly useful in analyzing two-symbol time series of relatively short length where the probabilities of occurrence of the two symbols are not equal. By inferring the universality class of an artificial-neural-network-based hybrid spin model through the value of the critical exponent δ, it is shown that for such time series, the proposed method provides a unique way to expose the real dynamics of the underlying complex system, in contrast to the analysis of waiting times in the time domain that leads to an ambiguous quantitative result.

Prime values of a sparse polynomial sequence

A distinguishing feature of certain intractable problems in prime number theory is the sparsity of the underlying sequence. Motivated by the general problem of finding primes in sparse polynomial sequences, we give an estimate for the number of primes of the shape x3+2y3 where y is small.

Asymptotic Counting in Conformal Dynamical Systems

In this monograph we consider the general setting of conformal graph directed Markov systems modeled by countable state symbolic subshifts of finite type. We deal with two classes of such systems: attracting and parabolic. The latter being treated by means of the former.We prove fairly complete asymptotic counting results for multipliers and diameters associated with preimages or periodic orbits ordered by a natural geometric weighting. We also prove the corresponding Central Limit Theorems describing the further features of the distribution of their weights.These results have direct applications to a wide variety of examples, including the case of Apollonian Circle Packings, Apollonian Triangle, expanding and parabolic rational functions, Farey maps, continued fractions, Mannenville-Pomeau maps, Schottky groups, Fuchsian groups, and many more. This gives a unified approach which both recovers known results and proves new results.Our new approach is founded on spectral properties of complexified Ruelle–Perron–Frobenius operators and Tauberian theorems as used in classical problems of prime number theory.

Multi-Radio Based Rendezvous Technique for Heterogeneous Cognitive Radio Sensor Network.

In a distributed cognitive radio (CR) sensor network, transmission and reception on vacant channels require cognitive radio nodes to achieve rendezvous. Because of the lack of adequate assistance from the network environment, such as the central controller and other nodes, assisted rendezvous for distributed CR is inefficient in a dynamic network. As a result, non-assisted blind rendezvous, which is unaware of its counterpart node, has recently led to a lot of interest in the research arena. In this paper, we study a channel rendezvous method based on prime number theory and propose a new multi-radio-based technique for non-assisted rendezvous with the blind and heterogeneous condition. The required time and the optimal number of radios for the guaranteed rendezvous are calculated using probability-based measurement. Analytical expressions for probabilistic guaranteed rendezvous conditions are derived and verified by Monte Carlo simulation. In addition, the maximum time to rendezvous (MTTR) is derived in closed form using statistical and probabilistic analysis. Under different channel conditions, our proposed solution leads to a substantial time reduction for guaranteed rendezvous. For the sake of over-performance of our proposed system, the simulation outcome is compared to a recently proposed heterogeneous and blind rendezvous method. The Matlab simulation results show that our proposed system’s MTTR gains range from to over for various parametric values of the system model.

A watermarking scheme using intellectual encoding-encryption based blind reversible integer wavelet-singular value decomposition transform for authenticity detection

Image processing plays vital role in analysis, storage and transmission of medical information. Digital watermarking (DWM) is considered as one of the more robust technique to protect patients’ Medical Health Record (MHR) in the field of telemedicine against various malicious attacks. In this paper, a blind reversible Integer Wavelet-Singular Value Decomposition (IWT-SVD) transform watermarking technique based on Encoded-encrypted Watermark is proposed for security authentication. Prior to watermarking, a novel encoding algorithm based on prime number theory with RSA encryption is used and followed by the QR code generation; is embedded into the patient health record. In this paper, different watermark embedding algorithms are discussed and a blind reversible IWT-SVD transform based DWM with novel encoding algorithm based on prime number theory is proposed with the suitable embedding algorithm along with RSA encryption applied for PHR watermark to be embedded in medical images and medical videos. QR code is also used to encode encrypted data so as to ensure reduced data payload. The proposed Encoded-Encrypted based DWM is applied to the medical images Ultra sound image, MRI image & CT scan images and 2-D & 3D medical videos. The proposed method is tested for various key values and it has achieved a PSNR on an average 80 db, SSIM of 0.95, MSE of 0.007, NAE of 0.006, NCC of almost 1 and the time of evaluation is 22 to50 seconds. The proposed technique based on novel encoding algorithm is compared with Huffman, Arithmetic coding, and with some classical watermarking systems.

Preliminary Discussion on Several Problems Related to the Divisor Function

This paper is divided into three parts to discuss the divisor function. It mainly combines the high power sum of the divisor function to study the solution number of the polynomial with integer coefficients "g(x)=0(modq)", and to improve some of the conclusions in the second chapter of Additive Theory of Prime Numbers and prove the improved conclusions.

Fourier optimization and prime gaps

We investigate some extremal problems in Fourier analysis and their connection to a problem in prime number theory. In particular, we improve the current bounds for the largest possible gap between consecutive primes assuming the Riemann hypothesis.

An elementary heuristic for Hardy–Littlewood extended Goldbach’s conjecture

The goal of this paper is to describe an elementary combinatorial heuristic that predicts Hardy and Littlewood’s extended Goldbach’s conjecture. We examine common features of other heuristics in additive prime number theory, such as Cramér’s model and density-type arguments, both of which our heuristic draws from. Apart from the prime number theorem, our argument is entirely elementary, in the sense of not involving complex analysis. The idea is to model sums of two primes by a hypergeometric probability distribution, and then draw heuristic conclusions from its concentration behavior, which follows from Hoeffding-type bounds.

Investigating the Primes

The aim of this expository article is to introduce the reader to some of the fundamental milestones in the study of prime numbers across several centuries. Among the important developments in the study of prime numbers, we review the history of the prime number theorem, the Riemann zeta function (in relation to prime number theory), and some recent investigations into spacings between consecutive primes. We also present an important application of prime numbers in safe data transmission, namely the ‘RSA public key cryptosystem’.

Some new density theorems for Dirichlet $L$-functions

We prove some new log-free density theorems for zeros of Dirichlet L-functions (which accordingly are more sharp than earlier ones near to the boundary line of the critical strip). The results can be applied in several problems of prime number theory.

A Career in Engineering

During the summers (1951–1954) that I was a graduate student in “pure mathematics,” I worked in the Systems Engineering Section of the Glenn L. Martin Company. I began to notice the applicability of supposedly pure topics like prime number theory and finite field theory to problems in communications. My first major applied effort involved developing the theory of “Shift Register Sequences.” (My book with this title will soon see its third edition, with a third publisher.) Much of my work has been in response to practical questions I was asked, for which I had the necessary mathematical tools. These topics have included comma-free codes, Costas arrays, Tuscan squares, Golomb rulers, zero-sidelobe radar, etc. My shift register work has had the broadest impact: to cell phone signals, the GPS system, error-correcting codes, radar, cryptography, etc. I am fortunate to have lived long enough to get some recognition for my work.

M(x)=o(x) Estimates for Beurling numbers

In classical prime number theory there are several asymptotic formulas said to be “equivalent” to the PNT. One is the bound M(x)=o(x) for the sum function of the Moebius function. For Beurling generalized numbers, this estimate is not an unconditional consequence of the PNT. Here we give two conditions that yield the Beurling version of the M(x) bound, and examples illustrating failures when these conditions are not satisfied.