Representation and Generation of Prime and Coprime Numbers by Using Structured Algebraic Sums

This paper addresses the computational methods and challenges associated with prime number generation, a critical component in encryption algorithms for ensuring data security. The generation of prime numbers efficiently is a critical challenge in various domains, including cryptography, number theory, and computer science. The quest to find more effective algorithms for prime number generation is driven by the increasing demand for secure communication and data storage and the need for efficient algorithms to solve complex mathematical problems. Our goal is to address this challenge by presenting two novel algorithms for generating prime numbers: one that generates primes up to a given limit and another that generates primes within a specified range. These innovative algorithms are founded on the formulas of odd-composed numbers, allowing them to achieve remarkable performance improvements compared to existing prime number generation algorithms. Our comprehensive experimental results reveal that our proposed algorithms outperform well-established prime number generation algorithms such as Miller-Rabin, Sieve of Atkin, Sieve of Eratosthenes, and Sieve of Sundaram regarding mean execution time. More notably, our algorithms exhibit the unique ability to provide prime numbers from range to range with a commendable performance. This substantial enhancement in performance and adaptability can significantly impact the effectiveness of various applications that depend on prime numbers, from cryptographic systems to distributed computing. By providing an efficient and flexible method for generating prime numbers, our proposed algorithms can develop more secure and reliable communication systems, enable faster computations in number theory, and support advanced computer science and mathematics research.

Public-key cryptosystems base their security on well-known number-theoretic problems, such as factorisation of a given number n . Hence, prime number generation is an absolute requirement. Many prime number generation techniques have been proposed up-to-date, which differ mainly in terms of complexity, certainty and speed. Pocklington's theorem, if implemented, can guarantee the generation of a true prime. The proposed implementation exhibits low complexity at the expense of long execution time.

Generating large prime number is a time consuming problem. It is useful in key generation in network security. The problem of generating prime number is easy to parallelize. Use of high performance computing (HPC) facilities can be applied for getting faster results. Embarrassing parallel pattern is applied to categories of algorithms where concurrency is explicit. We have conducted experiments on sequential and parallel approaches of prime numbers generation. In this paper, our objective is to analyze time taken in generating large prime numbers using multicore cluster. The result could be reusable in estimating time required for key generation in cryptography.

This chapter introduces basic concepts of elementary number theory such as divisibility, greatest common divisor, and prime and composite numbers. The principal results are Theorem 1.2, which establishes the existence of the greatest common divisor of any two integers, and Theorem 1.10 (the fundamental theorem of arithmetic), which shows that every integer greater than 1 can be represented as a product of prime factors in only one way (apart from the order of the factors). Many of the proofs make use of the following property of integers.

People are used prime numbers in secret codes to secure the message and data. Prime numbers are essential in computer security problem to perform certain operations over the Internet. It can be used for encryption. The larger the prime number used, the intricate it is to decipher the code. So finding the “massive” prime number is a current problem. One of the most widely used applications of prime number in computing is the RSA encryption system. A prime sieve is a fast type of algorithm for finding prime numbers. Sieve of Eratosthenes, Sieve of Sundaram and Sieve of Atkins are the algorithms that can be used to generate prime numbers. This research paper is to study and testing to find out, which algorithm is better used for large prime numbers in terms of time complexity. The prime sieve algorithm is coded in java language and the results are summarized.

Cell type classification and handling is a key issue for understanding biological systems. The advent of high multiplexing technologies increased the complexity of the classification process and new tools are needed to support the organization of this knowledge. Ipropose a classification based on both prime numbers and the fundamental theorem of arithmetic. As a not limiting example, I show the application of this method to unambiguously define any existing cell type using the CD nomenclature established by the Human Leukocyte Differentiation Antigens Workshops. This system allows for the unique identification of any possible combination of markers hence any cell population without previous knowledge and without the need to increment the system. This method can be the future basis of any database and ontology system dealing with cell types and beyond the biological field applies to the description of any entity characterized by a list of discrete qualities. © 2018 International Society for Advancement of Cytometry.

Public-key algorithms need prime numbers, any reasonably sized network needs lots of them. Before discussing the mathematics of prime number generation in the article the generation of the large random prime numbers is represented. A new algorithm of generating large prime numbers is also presented.

Preface. Notation. Chapter 1 Fundamentals. 1.0 Introduction. 1.1 A Famous Sequence of Numbers. 1.2 The Euclidean ALgorithm. The Oldest Algorithm. Reversing the Euclidean Algorithm. The Extended GCD Algorithm. The Fundamental Theorem of Arithmetic. Two Applications. 1.3 Modular Arithmetic. 1.4 Fast Powers. A Fast Alforithm for ExponentiationPowers of Matrices, Big-O Notation. Chapter 2 Congruences, Equations, and Powers. 2.0 Introduction. 2.1 Solving Linear Congruences. Linear Diophantine Equations in Two Variables. The Conductor. An Importatnt Quadratic Congruence. 2.2 The Chinese Remainder Theorem. 2.3 PowerMod Patterns. Fermat's Little Theorem. More Patterns in Powers. 2.4 Pseudoprimes. Using the Pseudoprime Test. Chapter 3 Euler's Function. 3.0 Introduction. 3.1 Euler's Function. 3.2 Perfect Numbers and Their Relatives. The Sum of Divisors Function. Perfect Numbers. Amicalbe, Abundant, and Deficient Numbers. 3.3 Euler's Theorem. 3.4 Primitive Roots for Primes. The order of an Integer. Primes Have PRimitive roots. Repeating Decimals. 3.5 Primitive Roots for COmposites. 3.6 The Universal Exponent. Universal Exponents. Power Towers. The Form of Carmichael Numbers. Chapter 4 Prime Numbers. 4.0 Introduction. 4.1 The Number of Primes. We'll Never Run Out of Primes. The Sieve of Eratosthenes. Chebyshev's Theorem and Bertrand's Postulate. 4.2 Prime Testing and Certification. Strong Pseudoprimes. Industrial-Grade Primes. Prime Certification Via Primitive Roots. An Improvement. Pratt Certificates. 4.3 Refinements and Other Directions. Other PRimality Tests. Strong Liars are Scarce. Finding the nth Prime. 4.4 A Doszen Prime Mysteries. Chapter 5 Some Applications. 5.0 Introduction. 5.1 Coding Secrets. Tossing a Coin into a Well. The RSA Cryptosystem. Digital Signatures. 5.2 The Yao Millionaire Problem. 5.3 Check Digits. Basic Check Digit Schemes. A Perfect Check Digit Method. Beyond Perfection: Correcting Errors. 5.4 Factoring Algorithms. Trial Division. Fermat's Algorithm. Pollard Rho. Pollard p-1. The Current Scene. Chapter 6 Quadratic Residues. 6.0 Introduction. 6.1 Pepin's Test. Quadratic Residues. Pepin's Test. Primes Congruent to 1 (Mod. 6.2 Proof of Quadratic Reciprocity. Gauss's Lemma. Proof of Quadratic Recipocity. Jacobi's Extension. An Application to Factoring. 6.3 Quadratic Equations. Chapter 7 Continuec Faction. 7.0 Introduction. 7.1 FInite COntinued Fractions. 7.2 Infinite Continued Fractions. 7.3 Periodic Continued Fractions. 7.4 Pell's Equation. 7.5 Archimedes and the Sun God's Cattle. Wurm's Version: Using Rectangular Bulls. The Real Cattle Problem. 7.6 Factoring via Continued Fractions. Chapter 8 Prime Testing with Lucas Sequences. 8.0 Introduction. 8.1 Divisibility Properties of Lucas Sequencese. 8.2 Prime Tests Using Lucas Sequencesse. Lucas Certification. The Lucas-Lehmer Algorithm Explained. Luca Pseudoprimes. Strong Quadratic Pseudoprimes. Primality Testing's Holy Grail. Chapter 9 Prime Imaginaries and Imaginary Primes. 9.0 Introduction. 9.1 Sums of Two Squares. 9.2 The Gaussian Intergers. Complex Number Theory. Gaussian Primes. The Moat Problem. The Gaussian Zoo. 9.3 Higher Reciprocity 325. Appendix A. Maathematica Basics. 1.0 Introduction. A.1 Plotting. A.2 Typesetting. Sending Files By E-Mail. A.3 Types of Functions. A.4 Lists. A.5 Programs. A.6 Solving Equations. A.7 Symbolic Algebra. Appendix B Lucas Certificates Exist. References. Index of Mathematica Objects. Subject Index.

Most children are aware that some numbers are even and the rest are odd, but they know little else about our number system. The “board-stretcher” model provides a means to discovering the idea of factors and obtaining a feeling for prime and composite numbers. It portrays the inverse relation between multiplication and division and, as a finale, it can be used to introduce the concept of multiplication by a fraction. This model suggests the notion that the sum of the prime factors of a number (except for four) is always less than the product of the factors. The Fundamental Theorem of Arithmetic (or Unique Factorization Theorem) is understood by youngsters when they see that every composite number can be expressed as a product of prime factors that is unique except for the order of the factors.

Recent IT research on prime numbers has focused on generating large primes forcryptography, neglecting the advancement of systematic generation of prime numbers insequence using techniques like sieving. Despite being relegated to educational contexts, sieving techniques hold untapped potential. Much of the existing work remains obscure to contemporaryresearchers, leaving numerous opportunities to enhance the field. This article aims to highlight the relevance of old sieving algorithms, demonstrating that modern approaches and creativity can yield significant progress. DOI: 10.61416/ceai.v26i4.9098

Background: The twin prime conjecture is regarded as a classic puzzle in number theory and one of the most well-known conjectures that has always perplexed us. Mathematician David Hilbert presented 23 key mathematical problems and conjectures to be solved at the International Congress of Mathematicians in 1900. He included the Bernhard Riemann conjecture, the Twin Prime Conjecture, and the Goldbach's conjecture in the eighth of 23 mathematical problems. Methods: Based on the "Differential Incremental Equilibrium Theory" [1], the infinite set of infinite prime numbers is divided, the increment equation of infinite prime numbers is established, and the tree-like set of prime numbers is obtained. Find the twin primes with a minimum unit [1\(\to\)1] of 2. Results: When a set of prime numbers is infinitely divided, there are 2[1\(\to\)1] pairs of prime numbers whose gap is equal to 2 and gap is not equal to 2. We gives a complete proof of the twin prime conjecture. It shows that the importance of "Differential Incremental Equilibrium Theory" [1] and infinite classification in twin prime conjecture. In a higher-level ideology, the set infinite partition classification confirms that the minimum unit is 2. It's a new way to prove Twin Prime Conjecture. Conclusion: This paper gives a complete proof of the establishment of the Twin Prime Conjecture.

In a long-term monitoring wireless sensor network (WSN) application, sensors are frequently deployed in a wide and an unattended geographical area to gather useful information for a long period of time. Although energy efficiency is affected by various factors, the wireless communication unit is typically the most energy-intensive component of wireless sensors. To extend the life of wireless sensors, they alternate between sleep and active modes to conserve energy. Thus, to exchange a message with neighboring sensors, both sending and receiving sensors must discover each other and stay awake simultaneously. This paper proposes a new neighbor discovery protocol (NDP) by enhancing U-Connect, a well-known protocol that constructs neighbor discovery schedules using only a single prime number. Although the proposed method shares the same characteristics as U-Connect, it offers greater flexibility than U-Connect in terms of duty cycles and schedule lengths. Our numerical analysis based on a power-latency (PL) product shows that the proposed method is more efficient than other NDPs such as Quorum, U-Connect, Disco, and ECNDP.

As smart mobile devices are widely used, mobile threats are more serious, so security in mobile becomes more and more important. However, the performance of these devices are not powerful enough to use the same security algorithms as PCs. Public key cryptosystems such as RSA need big primes to enhance the security, however, generating a big prime takes substantial time even on a PC. In this paper, we study two prime generation algorithms for smart mobile devices. First, we analyze a previous prime generation algorithm using a GCD test, named PGCD-MR, and show it sometimes performs inferior to the traditional TD-MR test. Second, we propose a new GCD test, named m-bit GCD-MR, for fast prime generation in both PCs and smart mobile devices. We compare the running times of PGCD-MR, m-bit GCD-MR, and TD-MR combinations on PCs and Samsung Galaxy Tab 10.1. The experimental results show our running time analysis is accurate (only 2% error) and m-bit GCD-MR test is the fastest among three prime generation algorithms. More exactly, m-bit GCD-MR test is about 20% faster than the TD-MR combination.

This review paper explores the critical role of number theory in shaping the foundations and advancements of modern cryptography and cybersecurity. As digital communication and data security become increasingly vital, the mathematical intricacies of number theory—such as prime number generation, modular arithmetic, and elliptic curves—form the backbone of secure encryption protocols like RSA, Diffie-Hellman, and Elliptic Curve Cryptography (ECC). The study employs a structured review methodology, drawing upon a wide range of scholarly publications and verified data up to 2014, including both global and Indian contexts. Key mathematical equations, comparative algorithmic performance, and numerical analyses have been integrated to assess the computational hardness and practical effectiveness of number theory-based cryptosystems. Results indicate that while traditional cryptographic schemes remain effective under classical computation, they face significant vulnerabilities in the face of emerging quantum technologies. The Indian mathematical tradition and its contribution to algorithmic logic further enrich the narrative, underscoring the need for indigenous innovation. This paper identifies major research gaps, particularly in the areas of post-quantum cryptography, algorithmic efficiency for low-resource environments, and provable security models. Future directions suggest hybrid mathematical frameworks and deeper exploration of algebraic structures to build resilient and scalable cryptographic systems. Overall, the paper establishes that number theory is not only a theoretical pursuit but a practical necessity in safeguarding digital infrastructure in an increasingly interconnected world.

We present an idea of curriculum development about prime numbers by using a real problem in mathematics.