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Journal articles on the topic 'Prime numbers theory'

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Published: 2 June 2025

Last updated: 31 July 2025

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1

Farkas, Gábor, Zsombor Kiss, Dániel Papatyi, and Krisztina Schäffer. "Prime Numbers." Mérnöki és Informatikai Megoldások, no. II. (October 20, 2020): 5–13. http://dx.doi.org/10.37775/eis.2020.2.1.

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"``Prime hunting" can be considered as a research area of computational number theory. Its goal is to find special combinations of integers and prove their primality. Four research groups, established by A. Járai between 1992 and 2014, published numerous world class scientific results. In this period, due to Járai's arithmetic routines fastest in the world, they reached the world record 19 times, namely found the largest known twin primes 9 times, Sophie Germain primes 7 times, a prime of the form n^4+1, a number which is simultaneously twin and Sophie Germain prime and the three largest known

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Chillali, Abdelhakim. "R-prime numbers of degree k." Boletim da Sociedade Paranaense de Matemática 38, no. 2 (2018): 75–82. http://dx.doi.org/10.5269/bspm.v38i2.38218.

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In computer science, a one-way function is a function that is easy to compute on every input, but hard to invert given the image of a random input. Here, "easy" and "hard" are to be understood in the sense of computational complexity theory, specifically the theory of polynomial time problems. Not being one-to-one is not considered sufficient of a function for it to be called one-way (see Theoretical Definition, in article). A twin prime is a prime number that has a prime gap of two, in other words, differs from another prime number by two, for example the twin prime pair (5,3). The question o

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KNOPFMACHER, ARNOLD, and FLORIAN LUCA. "ON PRIME-PERFECT NUMBERS." International Journal of Number Theory 07, no. 07 (2011): 1705–16. http://dx.doi.org/10.1142/s1793042111004447.

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We prove that the Diophantine equation [Formula: see text] has only finitely many positive integer solutions k, p1, …, pk, r1, …, rk, where p1, …, pk are distinct primes. If a positive integer n has prime factorization [Formula: see text], then [Formula: see text] represents the number of ordered factorizations of n into prime parts. Hence, solutions to the above Diophantine equation are designated as prime-perfect numbers.

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Oktaviani, Dinni Rahma, Muhammad Habiburrohman, and Fiki Syaban Nugroho. "ALTERNATIVE PROOF OF THE INFINITUDE PRIMES AND PRIME PROPERTIES." BAREKENG: Jurnal Ilmu Matematika dan Terapan 17, no. 1 (2023): 0475–80. http://dx.doi.org/10.30598/barekengvol17iss1pp0475-0480.

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Prime numbers is one of kind number that have many uses, one of which is cryptography. The uniqueness of prime numbers in their divisors and distributions causes prime numbers to be widely used in digital security systems. In number theory, one of famous theorem is Euclid theorem. Euclid theorem says about infinitely of prime numbers. Many alternative proof has been given by mathematician to find new theory or approximation of prime properties. The construction of proof give new idea about properties of prime number. So, in this study, we will give an alternative proof of Euclid theorem and in

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Budee U, Zaman. "New prime number theory." Annals of Mathematics and Physics 7, no. 2 (2024): 158–61. http://dx.doi.org/10.17352/amp.000119.

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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.

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Vinoo, Cameron. "-3 As the created root of all mathematics by numbers, Prime number 5 as the created template of all Prime number and pseudo-prime numbers(Mathematical Proof)." Journal of Progressive Research in Mathematics 14, no. 1 (2018): 2318–23. https://doi.org/10.5281/zenodo.3981236.

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The author has published several papers with JPRM which were unorthodox, but led to the acceptance of a major book on created mathematics , this small paper validates JPRM, and is a challenge to the entire current numbers theory if understood correctly .This small paper is the basic proof of the base of numbers in the created mathematics of the cone of Pythagoras 1:3 as Published at JPRM  , with the spiral arrangement of the Prime numbers and their multiples by the template of prime number 5, as the basis as shown separately in an upcoming book on created mathematics. The table entered in

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Mahato, Prabhat, and Aayush Shah. "A Review of Prime Numbers, Squaring Prime Pattern, Different Types of Primes and Prime Factorization Analysis." International Journal for Research in Applied Science and Engineering Technology 11, no. 7 (2023): 2036–43. http://dx.doi.org/10.22214/ijraset.2023.54904.

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Abstract: The study of prime numbers and their properties has always been an intriguing and fascinating topic for mathematicians. Primes can be considered the “basic building blocks,” the atoms, of the natural numbers. They play a significant role in number theory. Also, prime numbers, in this current world of computers and digitalization, have paramount significance for the computer programmers and scientists to tackle relevant real-life problems. Since long time, many studies and researches have been conducted regarding prime numbers pattern. In this paper, a squaring prime pattern is presen

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Eshmuminova, Yulduz. "BREAKING DOWN MOMENTS: SUM REPRESENTATION VIA ODD PRIME NUMBERS." International journal of mathematics and statistics 04, no. 01 (2024): 20–26. http://dx.doi.org/10.55640/ijmse-04-01-01.

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This article explores the mathematical concept of representing positive integers as sums of distinct odd prime numbers. The focus is on understanding the unique combinations and properties that arise when expressing numbers through this framework. Utilizing advanced combinatorial techniques and number theory principles, the study provides a comprehensive analysis of the conditions under which such representations are possible. We delve into the role of prime gaps, the frequency of prime numbers, and their influence on the sum decompositions of integers. Several novel findings are presented, in

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K.L, Bajaj. "Exploring Number Theory: From Prime Numbers to Cryptographic Algorithms." Modern Dynamics: Mathematical Progressions 1, no. 3 (2024): 10–14. https://doi.org/10.36676/mdmp.v1.i3.36.

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Number theory, a branch of pure mathematics devoted to the study of integers and integer-valued functions, has profound implications in various fields, particularly in cryptography. This paper delves into the intricate world of number theory, tracing its historical development and highlighting its pivotal role in modern cryptographic algorithms. We begin by exploring fundamental concepts such as prime numbers, greatest common divisors, and modular arithmetic, which form the bedrock of number theory. The significance of prime numbers is underscored by their application in key cryptographic meth

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Gueye, Ibrahima. "Conjecture in Additive Twin Primes Numbers Theory." Bulletin of Society for Mathematical Services and Standards 5 (March 2013): 27–30. http://dx.doi.org/10.18052/www.scipress.com/bsmass.5.27.

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For two millennia, the prime numbers have continued to fascinate mathematicians. Indeed, a conjecture which dates back to this period states that the number of twin primes is infinite. In 1949 Clement showed a theorem on twin primes. For the record, the theorem of Clement has quickly been known to be ineffective in the development of twin primes because of the factorial. This is why I thought ofusing the additive theory of numbers to find pairs of twin primes from the first two pairs of twin primes. What I have formulated as a conjecture. In same time i presentmy idea about the solution of the

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Sukhotin, A., and M. Zvyagin. "Alternative analysis: the prime numbers theory and an extension of the real numbers set." Bulletin of Science and Practice 398, no. 10(11) (2016): 10–14. https://doi.org/10.5281/zenodo.160909.

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Here we consider the theory of prime numbers at a new methodology. The theory of prime numbers is one of the most ancient mathematical branches. We found an estimate of the all prime numbers sum using the notions of infinite lager numbers and infinitely small numbers, farther we estimated the value of the maximal prime number. We proved that Hardy–Littlewood Hypothesis has the positive decision too. The infinite small numbers define a new methodology of the well–known function o(x) application. We use the sets of the theory of prime numbers and infinitely small numbers with a linear function t

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Lebowitz-Lockard, Noah. "Additively unique sets of prime numbers." International Journal of Number Theory 14, no. 10 (2018): 2757–65. http://dx.doi.org/10.1142/s179304211850166x.

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Spiro proved that the identity function is the only multiplicative function with [Formula: see text] for some prime [Formula: see text] and [Formula: see text] for all prime [Formula: see text] and [Formula: see text]. We determine the sets [Formula: see text] of primes for which restricting our condition to [Formula: see text] for all [Formula: see text] still implies that [Formula: see text] is the identity function. We prove that [Formula: see text] satisfies these conditions if and only if [Formula: see text] contains every prime that is not the larger element of a twin prime pair and [For

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Korniłowicz, Artur, and Dariusz Surowik. "Elementary Number Theory Problems. Part II." Formalized Mathematics 29, no. 1 (2021): 63–68. http://dx.doi.org/10.2478/forma-2021-0006.

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Summary In this paper problems 14, 15, 29, 30, 34, 78, 83, 97, and 116 from [6] are formalized, using the Mizar formalism [1], [2], [3]. Some properties related to the divisibility of prime numbers were proved. It has been shown that the equation of the form p 2 + 1 = q 2 + r 2, where p, q, r are prime numbers, has at least four solutions and it has been proved that at least five primes can be represented as the sum of two fourth powers of integers. We also proved that for at least one positive integer, the sum of the fourth powers of this number and its successor is a composite number. And fi

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Agdgomelashvili, Zurab. "Some interesting tasks from the classical number theory." Works of Georgian Technical University, no. 4(518) (December 15, 2020): 150–88. http://dx.doi.org/10.36073/1512-0996-2020-4-150-188.

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The article considers the following issues: – It’s of great interest for p and q primes to determine the number of those prime number divisors of a number 1 1 pq A p    that are less than p. With this purpose we have considered: Theorem 1. Let’s p and q are odd prime numbers and p  2q 1. Then from various individual divisors of the 1 1 pq A p    number, only one of them is less than p. A has at least two different simple divisors; Theorem 2. Let’s p and q are odd prime numbers and p  2q 1. Then all prime divisors of the number 1 1 pq A p    are greater than p; Theorem 3. Let’s q i

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Ezz-Eldien, Amal, Mohamed Ezz, Amjad Alsirhani, et al. "Computational challenges and solutions: Prime number generation for enhanced data security." PLOS ONE 19, no. 11 (2024): e0311782. http://dx.doi.org/10.1371/journal.pone.0311782.

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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 presen

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Perucca, Antonella, and Pietro Sgobba. "Kummer Theory for Number Fields and the Reductions of Algebraic Numbers II." Uniform distribution theory 15, no. 1 (2020): 75–92. http://dx.doi.org/10.2478/udt-2020-0004.

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AbstractLet K be a number field, and let G be a finitely generated and torsion-free subgroup of K×. For almost all primes p of K, we consider the order of the cyclic group (G mod 𝔭), and ask whether this number lies in a given arithmetic progression. We prove that the density of primes for which the condition holds is, under some general assumptions, a computable rational number which is strictly positive. We have also discovered the following equidistribution property: if ℓe is a prime power and a is a multiple of ℓ (and a is a multiple of 4 if ℓ =2), then the density of primes 𝔭 of K such th

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Diego, Real. "Open path theory: Pattern and structure in prime numbers." Annals of Mathematics and Physics 6, no. 2 (2023): 141–48. http://dx.doi.org/10.17352/amp.000093.

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The Open Path theory, supported by experimental data, is presented. The main hypothesis proposes that Prime Numbers's positions are determined by previous Prime Numbers as well as their spacing, in a complex, but deterministic way. The concepts of Open Path, Perfect Space, and Primorial Perfect Space are introduced. The Open Path theory can predict prime gaps of any minimum predetermined size. Two rudimentary algorithms based on this theory are presented. The first algorithm returns a sample (a few hundredth of numbers) containing 25 % of Prime Numbers at distances above 1011. The mirrored sam

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Sankei, Daniel, Loyford Njagi, and Josephine Mutembei. "A Logical Proof of the Polignac's Conjecture Based on Partitions of an Even Number of a New Formulation." Asian Research Journal of Mathematics 20, no. 3 (2024): 1–11. http://dx.doi.org/10.9734/arjom/2024/v20i3787.

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Polignac's Conjecture, proposed by Alphonse de Polignac in the 19th century, is a captivating hypothesis that extends the notion of twin primes to a broader context. It posits that for any even positive integer 2k, there exist infinitely many pairs of consecutive prime numbers whose difference is 2k. This conjecture is a natural generalization of the Twin Prime Conjecture, which focuses solely on pairs of primes differing by two. The conjecture has significant implications for our understanding of the distribution of prime numbers and the nature of their gaps and its exploration serves as a te

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Mosahab, Arya. "Exploring the Infinitude of Primes and Related Conjectures." European Journal of Mathematics and Statistics 6, no. 3 (2025): 1–5. https://doi.org/10.24018/ejmath.2025.6.3.399.

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This paper presents several classical and modern proofs demonstrating the infinitude of prime numbers. The discussion includes Euclid’s original argument as well as alternative approaches including analytic, topological, and combinatorial proofs. In addition, this paper discusses two open problems in number theory: the infinitude of Mersenne primes and the Twin Prime Conjecture. The aim is to provide an overview of both established results and ongoing challenges in the study of prime numbers.

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Honwei, Shi, Mi Zhou, Zhang Delong, Jiang Xingyi, and He Songting. "Even numbers are the sum of two prime numbers." Greener Journal of Science, Engineering and Technological Research 9, no. 1 (2019): 8–11. https://doi.org/10.15580/GJSETR.2019.1.040919068.

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<strong>"Even numbers are the sum of two prime numbers." This is the description of Goldbach's conjecture, and in Canadian Gaye's book "unsolved problems in number theory", it is an open question to put forward a contrary conjecture that "even numbers are the difference between two prime numbers". In this paper, Chandra sieve is used to deduce that the sum of large and even numbers is the sum of two prime numbers, and that "even numbers are the difference between two prime numbers" is a great possibility. At the same time, it is possible to guess

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Korniłowicz, Artur, and Adam Naumowicz. "Elementary Number Theory Problems. Part V." Formalized Mathematics 30, no. 3 (2022): 229–34. http://dx.doi.org/10.2478/forma-2022-0018.

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Summary This paper reports on the formalization of ten selected problems from W. Sierpinski’s book “250 Problems in Elementary Number Theory” [5] using the Mizar system [4], [1], [2]. Problems 12, 13, 31, 32, 33, 35 and 40 belong to the chapter devoted to the divisibility of numbers, problem 47 concerns relatively prime numbers, whereas problems 76 and 79 are taken from the chapter on prime and composite numbers.

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Nicolás, José Alfonso López. "Advances in additive number theory." Boletim da Sociedade Paranaense de Matemática 41 (December 26, 2022): 1–20. http://dx.doi.org/10.5269/bspm.51233.

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We obtain sufficient conditions to know if given a positive even integer number and a set of positive integer numbers being all even or all odd, such a number can be expressed as sum of two elements of this set. As consequence we obtain a result which, when applied to the prime numbers set, would prove Goldbach's Conjecture provided that certain conditions are satisfied. These hypothesis include Prime Consecutive Conjecture, which is a generalized form of Twin Prime Conjecture. In addition, we extend these results to sets of positive real numbers, even for two different sets. We also obtain a

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Ibrahim, Mohammed Ali Faya, and Alwah Saleh Ahmad Alqarni. "SEMI TWIN PRIME NUMBERS." JP Journal of Algebra, Number Theory and Applications 45, no. 2 (2020): 205–30. http://dx.doi.org/10.17654/nt045020205.

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KAPLAN, ITAY, and SAHARON SHELAH. "DECIDABILITY AND CLASSIFICATION OF THE THEORY OF INTEGERS WITH PRIMES." Journal of Symbolic Logic 82, no. 3 (2017): 1041–50. http://dx.doi.org/10.1017/jsl.2017.16.

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AbstractWe show that under Dickson’s conjecture about the distribution of primes in the natural numbers, the theory Th (ℤ , +, 1, 0, Pr) where Pr is a predicate for the prime numbers and their negations is decidable, unstable, and supersimple. This is in contrast with Th (ℤ , +, 0, Pr, <) which is known to be undecidable by the works of Jockusch, Bateman, and Woods.

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Khalil, Lolav Ahmed. "Distribution of the prime numbers." Bulletin of Applied Mathematics and Mathematics Education 4, no. 1 (2024): 19–26. http://dx.doi.org/10.12928/bamme.v4i1.10408.

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This research explores the distribution of prime numbers, which are a fundamental topic in number theory. The study originated from the author's fascination with mathematics and the desire to discover something novel. The research proposes that the distribution of prime numbers follows a regular pattern starting from the number 2. The author suggests that prime numbers can be obtained by dividing certain even numbers that have four factors by the number 2, resulting in prime numbers in sequential order. This hypothesis was tested and confirmed through the practical application of the proposed

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Khusid, Mykhaylo. "Solving Two Problems IN Number Theory." European Journal of Mathematics and Statistics 2, no. 3 (2021): 8–9. http://dx.doi.org/10.24018/ejmath.2021.2.3.24.

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In 1995, Olivier Ramaret proved that any even number is the sum of no more than 6 primes. From the validity of Goldbach's ternary hypothesis (proved in 2013 year) it follows that any even number is the sum of not more than 4 numbers [1]. In the article, the author confirms the above and proves that the cause and effect of this is any even number the sum of not more than two prime and twin primes are infinite [8]-[14].

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Khalil, Lolav Ahmed. "On the Distribution of the Prime Numbers." Journal of Basic & Applied Sciences 20 (December 12, 2024): 182–89. https://doi.org/10.29169/1927-5129.2024.20.17.

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This research explores the distribution of prime numbers, which are a fundamental topic in number theory. The study originated from the author's fascination with mathematics and the desire to discover something novel. The research proposes that the distribution of prime numbers follows a regular pattern starting from the number 2. The author suggests that prime numbers can be obtained by dividing certain even numbers that have four factors by the number 2, resulting in prime numbers in sequential order. This hypothesis was tested and confirmed through the practical application of the proposed

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Lattanzi, Daniele. "Computer Simulation Model of Prime Numbers." Journal of Advances in Mathematics and Computer Science 38, no. 8 (2023): 101–21. http://dx.doi.org/10.9734/jamcs/2023/v38i81794.

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Prime numbers represent one of the major open problems in number theory mostly in that at present it is not possible to state that the induction principle holds for them. The methodology of experimental mathematics has been little endeavored in this field thus the present report deals with an innovative approach to the problem of primes treated as raw experimental data and as elements of larger and larger finite sequences {Pn}. The modified Chi-square function in the form -1/X2k(A,n/μ) with the ad-hoc A, k and μ parameters is the best-fit function of the finite sequences of primes {Pn}, like t

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Spahn, Mark, Ron Lancaster, Deborah Moore-Russo, and Gerald Rising. "An unexpected use of primes: solving sudokus by calculator." Mathematical Gazette 94, no. 530 (2010): 224–32. http://dx.doi.org/10.1017/s0025557200006483.

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This essay demonstrates an application of prime numbers to the development of a calculator program that solves sudoku puzzles. Among the positive integers, the primes—numbers with exactly two divisors, the numbers themselves and 1—are central to our thinking about numbers. They give us a basis for factoring and divisibility and they contribute to the solution of many problems in the mathematical field of number theory. More important for the purposes of this paper, they provide a way of representing numbers uniquely by prime factors. For example, 6221592 = 23 × 32 × 13 × 172 × 23, any other fa

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SANCHIS-LOZANO, MIGUEL-ANGEL, J. FERNANDO BARBERO G., and JOSÉ NAVARRO-SALAS. "PRIME NUMBERS, QUANTUM FIELD THEORY AND THE GOLDBACH CONJECTURE." International Journal of Modern Physics A 27, no. 23 (2012): 1250136. http://dx.doi.org/10.1142/s0217751x12501369.

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Motivated by the Goldbach conjecture in number theory and the Abelian bosonization mechanism on a cylindrical two-dimensional space–time, we study the reconstruction of a real scalar field as a product of two real fermion (so-called prime) fields whose Fourier expansion exclusively contains prime modes. We undertake the canonical quantization of such prime fields and construct the corresponding Fock space by introducing creation operators [Formula: see text] — labeled by prime numbers p — acting on the vacuum. The analysis of our model, based on the standard rules of quantum field theory and t

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Jiang, Zirui. "Factoring primes and sums of two squares." Theoretical and Natural Science 13, no. 1 (2023): 18–22. http://dx.doi.org/10.54254/2753-8818/13/20240744.

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Mathematicians began to study a series of properties about numbers a long time ago, and a new field of mathematics, the number theory, was born from this. Some special properties of numbers in the number theory make mathematicians use the knowledge of group theory to make some ingenious answers when considering some problems. In the analytic number theory, equations related to numbers have always been a concern of mathematicians. The most famous Fermat's last theorem also brought long-term troubles to countless mathematicians and was finally proved by the British mathematician Wiles. Many famo

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Fulara, Bhawana, Arvind Bhatt, Deepak Kumar Sharma, Shubham Agarwal, Geeta Mathpal, and Rajesh Mathpal. "Statistical Analysis and Distribution of Fermat Pseudoprimes Within the Given Interval." International Journal of Experimental Research and Review 44 (October 30, 2024): 115–20. http://dx.doi.org/10.52756/ijerr.2024.v44spl.010.

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Prime numbers are natural numbers that can only be divided by one and the original number. There is more than one of them. Error-correcting codes used in telecommunications are generated using prime numbers. They guarantee automatic message correction both during transmission and reception. Algorithms used in public-key cryptography are built upon primes. They're also employed in the production of pseudorandom numbers. Mathematicians and many other scientific and technological communities have always been fascinated by prime numbers. Additionally, computer engineers can use it to tackle a wide

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Vavilov, Nikolai. "Computers as Novel Mathematical Reality. VI. Fermat numbers and their relatives." Computer tools in education, no. 4 (December 28, 2022): 5–68. http://dx.doi.org/10.32603/2071-2340-2022-4-5-67.

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In this part, which constitutes a pendent to the part dedicated to Mersenne numbers, I continue to discuss the fantastic contributions towards the solution o classical problems of number theory achieved over the last decades with the use of computers. Specifically, I address primality testing, factorisations and the search of prime divisors of the numbers of certain special form, primarily Fermat numbers, their friends and relations, such as generalised Fermat numbers, Proth numbers, and the like. Furthermore, we discuss the role of Fermat primes and Pierpoint primes in cyclotomy.

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Sárközy, A., and C. L. Stewart. "On exponential sums over prime numbers." Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics 46, no. 3 (1989): 423–37. http://dx.doi.org/10.1017/s1446788700030913.

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AbstractIn this article we establish an estimate for a sum over primes that is the analogue of an estimate for a sum over consecutive integers which has proved to be very useful in applications of exponential sums to problems in number theory.

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Abdel-Mageed, M., Ahmed Salim, Walid Osamy, and Ahmed M. Khedr. "A Study on the Statistical Properties of the Prime Numbers Using the Classical and Superstatistical Random Matrix Theories." Advances in Mathematical Physics 2021 (September 21, 2021): 1–17. http://dx.doi.org/10.1155/2021/9956518.

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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 fr

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Sankei, Daniel, Loyford Njagi, and Josephine Mutembei. "Application of Odd Pairs of Partitions of an Even Number of a New Formulation in Validating the Twin Prime Conjecture." Journal of Advances in Mathematics and Computer Science 39, no. 9 (2024): 40–45. http://dx.doi.org/10.9734/jamcs/2024/v39i91925.

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The Twin Prime Conjecture posits the existence of infinitely many pairs of prime numbers (p, p + 2), where both p and p + 2 are prime. Despite centuries of investigation, a definitive proof remains elusive. Prime numbers, defined by their indivisibility except by one and themselves, display an apparently erratic distribution. Researchers have utilized a combination of theoretical insights, computational analysis, and innovative mathematical techniques in their quest to solve this conjecture. However, the unpredictable nature of prime occurrences has kept this problem open in Number Theory. Thi

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Sankei, Daniel, Loyford Njagi, and Josephine Mutembei. "A Detailed Proof of the Strong Goldbach Conjecture Based on Partitions of a New Formulation of a Set of Even Numbers." Asian Research Journal of Mathematics 20, no. 4 (2024): 8–17. http://dx.doi.org/10.9734/arjom/2024/v20i4793.

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The Strong Goldbach's conjecture, a fundamental problem in Number Theory, asserts that every even integer greater than 2 can be expressed as the sum of two prime numbers. Despite significant efforts over centuries, this conjecture remains unproven, challenging the core of mathematics. The known algorithms for attempting to prove or verify the conjecture on a given interval [a,b] consist of finding two sets of primes Pi and Pj such that Pi+Pj cover all the even numbers in the interval [a,b]. However, the traditional definition of an even number as 2n for n ∈ ℕ (where ℕ is the set of natural num

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Gueye, Ibrahima. "Polynomial Characterization of Twin Primes in Function of another Prime." Bulletin of Society for Mathematical Services and Standards 5 (March 2013): 37–39. http://dx.doi.org/10.18052/www.scipress.com/bsmass.5.37.

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For two millennia, the prime numbers have continued to fascinatemathematicians. Indeed, a conjecture which dates back to this period states that thenumber of twin primes is infinite. In 1949 Clement showed a theorem on twin primesIn this paper I give the proof of a polynomial characterization of twin primes usingadditive primes number theory.

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Burkhart, Jerry. "Building Numbers from Primes." Mathematics Teaching in the Middle School 15, no. 3 (2009): 156–67. http://dx.doi.org/10.5951/mtms.15.3.0156.

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Use building blocks to create a visual model for prime factorizations. Students can explore many concepts of number theory, including the relationship between greatest common factors and least common multiples.

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Samarawickrama, Mahendra. "Consciousness and Mathematics: A Number Theoretic Approach to Modelling Reality." Journal of Physics: Conference Series 3027, no. 1 (2025): 012013. https://doi.org/10.1088/1742-6596/3027/1/012013.

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Abstract This research analyses the fundamentals of numbers for interpreting consciousness and reality. Complementing Gödel’s incompleteness theorems, we adopted number theory to explore non-referential and self-referential constructs. By examining consciousness, causation, and fundamental mathematical models of reality, we analysed stages of cognition and the emergence of self-referencing as a limitation, which brings incompleteness and undecidability to a framework. By postulating prime numbers as a non-referential fundamental basis, the study underscores their critical role in forming a com

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Broughan, Kevin A. "Adic Topologies for the Rational Integers." Canadian Journal of Mathematics 55, no. 4 (2003): 711–23. http://dx.doi.org/10.4153/cjm-2003-030-3.

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AbstractA topology on ℤ, which gives a nice proof that the set of prime integers is infinite, is characterised and examined. It is found to be homeomorphic to ℚ, with a compact completion homeomorphic to the Cantor set. It has a natural place in a family of topologies on ℤ, which includes the p-adics, and one in which the set of rational primes ℙ is dense. Examples from number theory are given, including the primes and squares, Fermat numbers, Fibonacci numbers and k-free numbers.

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HASSANI, MEHDI. "ON THE RATIO OF THE ARITHMETIC AND GEOMETRIC MEANS OF THE PRIME NUMBERS AND THE NUMBER e." International Journal of Number Theory 09, no. 06 (2013): 1593–603. http://dx.doi.org/10.1142/s1793042113500450.

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We study the asymptotic behavior of the sequence with general term consisting of the ratio An by Gn, the arithmetic and geometric means of the prime numbers p1, p2, …, pn, respectively, in which, pn denotes the nth prime number.

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Shcherban, V. "Ultra fast finding all the prime numbers: formula." Bulletin of Science and Practice, no. 9 (September 15, 2017): 8–13. https://doi.org/10.5281/zenodo.891161.

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Finding very large prime numbers is still considered a hard work. Existing algorithms already employ splitting numbers into simple multipliers, which exceed <em>10<sup>110</sup>.</em> This well takes 24 hours of the world’s most powerful ECM. Now we shall prove the opposite: no algorithms of random number primality are needed. Not a continuous work of a medium–power computer and the result is ready. The large prime numbers make the basis for protection of electronic commerce and electronic post. As some of the intruders gradually manage to compute them, knowing cryptologists keep renewing inve

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Erdős, Paul, Péter Kiss, and Carl Pomerance. "On prime divisors of Mersenne numbers." Acta Arithmetica 57, no. 3 (1991): 267–81. http://dx.doi.org/10.4064/aa-57-3-267-281.

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Liu, Hong-Quan, and Jie Wu. "Numbers with a large prime factor." Acta Arithmetica 89, no. 2 (1999): 163–87. http://dx.doi.org/10.4064/aa-89-2-163-187.

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Chakraborty, Kalyan, Florian Luca, and Anirban Mukhopadhyay. "Class numbers with many prime factors." Journal of Number Theory 128, no. 9 (2008): 2559–72. http://dx.doi.org/10.1016/j.jnt.2008.03.010.

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SAVIN, DIANA, and MIRELA ŞTEFĂNESCU. "A NECESSARY CONDITION FOR CERTAIN PRIMES TO BE WRITTEN IN THE FORM xq + ryq." Journal of Algebra and Its Applications 10, no. 03 (2011): 435–43. http://dx.doi.org/10.1142/s0219498811004665.

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Letnan Kolonel Elektronika Imat Rakhmat Hidayat, S.T., M.Eng. "SAVINGS TIME EXECUTION PRIMA NUMBERS GENERATOR USING BIT-ARRAY STRUCTURE." MULTICA SCIENCE AND TECHNOLOGY (MST) 1, no. 1 (2021): 20–27. http://dx.doi.org/10.47002/mst.v1i1.202.

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Prime number in growth computer science of number theory and very need to yield an tool which can yield an hardware storey level effectiveness use efficiency and Existing Tools can be used to awaken regular prime number sequence pattern, structure bit-array represent containing subdividing variables method of data aggregate with every data element which have type of equal, and also can be used in moth-balls the yielded number sequence. Prime number very useful to be applied by as bases from algorithm kriptografi key public creation, hash table, best algorithm if applied hence is prime number i

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Brito da Silva, Hamilton. "New properties of divisors of natural number." Notes on Number Theory and Discrete Mathematics 29, no. 2 (2023): 276–83. http://dx.doi.org/10.7546/nntdm.2023.29.2.276-283.

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The divisors of a natural number are very important for several areas of mathematics, representing a promising field in number theory. This work sought to analyze new relations involving the divisors of natural numbers, extending them to prime numbers. These are relations that may have an interesting application for counting the number of divisors of any natural number and understanding the behavior of prime numbers. They are not a primality test, but they can be a possible tool for this and could also be useful for understanding the Riemann’s zeta function that is strongly linked to the distr

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Mykhalevych, Volodymyr, and Leonid Maidanevych. "USE OF THE MAPLE SYSTEM IN MATHEMATICAL PROBLEMS OF CRYPTOGRAPHY. PART 1. ELEMENTARY THEORY OF NUMBERS." Information technology and computer engineering 59, no. 1 (2024): 105–18. http://dx.doi.org/10.31649/1999-9941-2024-59-1-105-118.

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On the basis of the analysis of literary sources, a conclusion was made about the relevance of using the environment of the Maple computer mathematics system for the purpose of creating software for conducting scientific research and creating educational and methodological materials for solving typical mathematical problems of cryptography. It is noted that the most famous and widespread cryptographic algorithm with a public key RSA is based on a number of problems of elementary number theory that can be solved using standard tools of the Maple system. This work examines the specified standard

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