Sieve Of Eratosthenes Research Articles

Reducing the space used by the sieve of Eratosthenes when factoring

We present a version of the sieve of Eratosthenes that can factor all integers ≤x in O(xlog⁡log⁡x) arithmetic operations using at most O(x/log⁡log⁡x) bits of space. Among algorithms that take the optimal O(xlog⁡log⁡x) time, this new space bound is an improvement of a factor proportional to log⁡xlog⁡log⁡x over the implied previous bound of O(xlog⁡x). We also show our algorithm performs well in practice.

Distance between consecutive elements of the multiplicative group of integers modulo n

For a prime number $p$, we consider its primorial $P:=p\#$ and $U(P):={\left(\ZZ{P}\right)}^\times$ the set of elements of the multiplicative group of integers modulo $P$ which we represent as points anticlockwise on a circle of perimeter $P$. These points considered with wrap around modulo $P$ are those not marked by the Eratosthenes sieve algorithm applied to all primes less than or equal to $p$. In this paper, we are mostly concerned with providing formulas to count the number of gaps of a given even length $D$ in $U(P)$ which we note $K(D,P)$. This work, presented with different notations is closely related to [5]. We prove the formulas in three steps. Although only the last step relates to the problem of gaps in the Eratosthenes sieve (see Section 3.2.2) the previous formulas may be of interest to study occurrences of defined gaps sequences. <ul> <li>For a positive integer $n$, we prove a general formula based on the inclusion-exclusion principle to count the number of occurrences of configurations<sup>1</sup> in any subset of $\ZZ{n}$. (see Equation (7) in Theorem 2.1).</li> <li>For a square-free integer $P$, we particularize this formula when the subset of interest is $U(P)$. (see Equation (11) in Theorem 3.2).</li> <li>For a prime $p$ and its primorial $P:=p\#$, we particularize the formula again to study gaps in $U(P)$. Given a positive integer $D$ representing a distance on the circle, we give formulas to count $K(D,P)$ the number of gaps of length $D$ between elements of $U(P)$ (see Equation (15) and Section 4.1).</li> </ul> In addition, we provide a formula (see Equation (27) in Theorem 5.1) to count the number of occurrences of gaps of an even length $N$ that contain exactly $i$ elements of $U(P)$. <sup>1</sup> <small>A defined sequence of gaps between the elements of the subset; this is referred to as a constellation in [5].</small>

Computational challenges and solutions: Prime number generation for enhanced data security.

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.

Exploring Polygonal Number Sieves through Computational Triangulation

The paper deals with the exploration of subsequences of polygonal numbers of different sides derived through step-by-step elimination of terms of the original sequences. Eliminations are based on special rules similarly to how the classic sieve of Eratosthenes was developed through the elimination of multiples of primes. These elementary number theory activities, appropriate for technology-enhanced secondary mathematics education courses, are supported by a spreadsheet, Wolfram Alpha, Maple, and the Online Encyclopedia of Integer Sequences. General formulas for subsequences of polygonal numbers referred to in the paper as polygonal number sieves of order k, that include base-two exponential functions of k, have been developed. Different problem-solving approaches to the derivation of such and other sieves based on the technology-immune/technology-enabled framework have been used. The accuracy of computations and mathematical reasoning is confirmed through the technique of computational triangulation enabled by using more than one digital tool. A few relevant excerpts from the history of mathematics are briefly featured.

Binary expansions of prime reciprocals

Prime numbers have been always of great interest. In this work, we explore the prime numbers from a sieve other than the Eratosthenes sieve. Given a prime number $p$, we consider the binary expansion of $\frac{1}{p}$ and, in particular, the size of the period of $\frac{1}{p}$. We show some results that relate the size of the period to properties of the prime numbers.

An inequality related to the sieve of Eratosthenes

Let Φ(x,y) denote the number of integers n∈[1,x] free of prime factors ≤y. We show that but for a few small cases, Φ(x,y)<.6x/log⁡y when y≤x.

The Sieve of Eratosthenes and a Partition of the Natural Numbers

The Sieve of Eratosthenes and a Partition of the Natural Numbers

On wide-output sieves

We describe and compare several novel sieve-like methods. They assign values of several functions (i.e., the prime omega functions ω and Ω and the divisor function d) to each natural number in the considered range of integer numbers. We prove that in some cases the algorithms presented have a relatively small computational complexity. A more detailed output is indeed obtained with respect to the original Sieve of Eratosthenes.

Numerical analysis approach to twin primes conjecture

The purpose of this paper is to demonstrate how the modified Sieve of Eratosthenes is used to have an approach to twin prime conjecture. If the Sieve is used in its basic form, it does not produce anything new. If it is used through the numerical analysis method explained in this paper, we obtain a specific counting of twin primes. This counting is based on the false assumption that distribution of primes follows punctually the Logarithmic Integral function denoted as Li(x) (see [5] and [10], pp. 174–176). It may be a starting point for future research based on this numerical analysis method technique that can be used in other mathematical branches.

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Using Sieve of Eratosthenes for the Factor Analysis of Neutrosophic Form of the Five Facet Mindfulness Questionnaire as an Alternative Confirmatory Factor Analysis

In this study, the Five Facet Mindfulness Questionnaire which was adapted from the short form of the Five Facet Mindfulness Questionnaire was evaluated and this scale into neutrosophic form was converted and the results of the scale were compared for proposing new type confirmatory analysis procedure as well as developing neutrosophic scales. The exploratory factor analysis was used in the analysis of the data. Besides, test results were analyzed for Kaiser–Meyer–Olkin and Bartlett values, common factor variance values, scree plot graphs, and the principal component analysis results. The sample of the study consists of 194 students in mathematics departments at Bitlis Eren University and I˘gdır University in Turkey by convenience sampling method. A convenience sampling is a kind of non-probability sampling procedure in which the sample is obtained from a group of individuals easily accessible or reachable. The convenience sampling method was chosen in this study because the study aims to examine the structure of the measurement tool rather than the psychological characteristics of a particular population. First of all, it is observed that if any classical scale can be converted into a neutrosophic one. It is observed that the sub-dimensions of a neutrosophic scale as agree, disagree, and undecided might not have a similar factor structure to the classical one. Interestingly, in the factor analysis of the neutrosophic scale, both classical and the agreement part of the neutrosophic scales have the same factors, implying that the one-dimensional classical scale measures the agreement degree of the participants. When the factor analysis was conducted to disagreement and vagueness dimensions, it seemed that some factors were eliminated and even some new factors emerged, indicating that in human cognition those three dimensions can be taken as independent of each other, just as assumed by neutrosophic logic. The another important implication of the factor analysis is that the neutrosophic forms of any questionnaire can be used for the validity of the classical ones. Loads of items or their accumulation into factors are compared to the classical scale and the three-dimensional neutrosophic scale in the factor, so that the corresponding ones in the same factors and the items or factors that do not correspond to each other are eliminated. It is very similar to the Sieve of Eratosthenes, which is an ancient algorithm for finding prime numbers up to any given limit where each prime is taken as an independent base or dimension and multiples of the selected primes in a given interval are eliminated until there are only prime numbers left. Finally, the reliability of three independent dimensions of the neutrosophic forms of any questionnaire can also be used to check whether the measurement tool is reliable. Low-reliability results in any dimensions may imply that the scale has some problems in terms of meaning, language, or other factors.

Mathematical Modelling of COVID-19 and Solving Riemann Hypothesis, Polignac&amp;#39;s and Twin Prime Conjectures Using Novel Fic-Fac Ratio With Manifestations of Chaos-Fractal Phenomena

COVID-19 originated from Wuhan, China in December 2019. Declared by the World Health Organization on March 11, 2020; COVID-19 pandemic has resulted in unprecedented negative global impacts on health and economy. International cooperation is required to combat this &amp;rdquo;Incompletely Predictable&amp;rdquo; pandemic. With manifestations of Chaos-Fractal phenomena, we mathematically model COVID-19 and solve [unconnected] open problems in Number theory using our versatile Fic-Fac Ratio. Computed as Information-based complexity, our innovative Information-complexity conservation constitutes a unique all-purpose analytic tool associated with Mathematics for Incompletely Predictable problems. These problems are literally &amp;rdquo;complex systems&amp;rdquo; containing well-defined Incompletely Predictable entities such as nontrivial zeros and two types of Gram points in Riemann zeta function (or its proxy Dirichlet eta function) together with prime and composite numbers from Sieve of Eratosthenes. Correct and complete mathematical arguments for first key step of converting this function into its continuous format version, and second key step of using our unique Dimension (2x -N) system instead of this Sieve result in primary spin-offs from first key step consisting of providing proof for Riemann hypothesis (and explaining closely related two types of Gram points), and second key step consisting of providing proofs for Polignac&amp;rsquo;s and Twin prime conjectures.

Comparison and Performance Evaluation of Prime Number Generation Using Sieves Algorithm

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.

Arithmetic functions of higher-order primes

The sieve of Eratosthenes (SoE) is a well-known method of extracting the set of prime numbers ℙ from the set positive integers ℕ. Applying the SoE again to the index of the prime numbers will result in the set of prime-indexed primes ℙ2={3,5,11,17,31,…}. More generally, the application of the SoE k-times will yield the set ℙk of k-th order primes. In this paper, we give an upper bound for the n-th k-order prime as well as some results relating to number-theoretic functions over ℙk.

An improved sieve of Eratosthenes

We show how to carry out a sieve of Eratosthenes up to N in space O(N^{1/3} (log N)^{2/3}) and essentially linear time. These bounds constitute an improvement over the usual versions of the sieve, which take space about O(sqrt{N}) and essentially linear time. We can also apply our sieve to any subinterval of [1,N] of length O(N^{1/3} (log N)^{2/3}) in time essentially linear on the length of the interval. Before, such a thing was possible only for subintervals of [1,N] of length >> sqrt{N}. Just as in (Galway, 2000), our approach is related to Diophantine approximation, and also has close ties to Voronoi's work on the Dirichlet divisor problem. The advantage of the method here resides in the fact that, because the method we will give is based on the sieve of Eratosthenes, we will also be able to use it to factor integers, and not just to produce lists of consecutive primes.

The Comparison of Methods for Generating Prime Numbers between The Sieve of Eratosthenes, Atkins, and Sundaram

Prime numbers are unique numbers. Prime numbers are numbers that only have a dividing factor consisting of numbers 1 and the number itself. The prime numbers from 1 to n that are relatively small can be generated manually, but a prime number generator algorithm is needed to generate prime numbers on a large scale. This study compares three prime number generator algorithms, namely: The Sieve of Eratosthenes, The Sieve of Atkins, and The Sieve of Sundaram. These three sieve algorithms have their own differences in generating prime numbers. The Sieve of Eratosthenes uses a simpler method by crossing multiples of prime numbers and marking them as non-prime numbers. The Sieve of Atkins uses several requirements for quadratic equations and modulus in determining prime numbers. The Sieve of Sundaram has an algorithm similar to The Sieve of Atkins, but there are requirements for linear equations to determine prime numbers. This study aims to see a comparison of these three algorithms in terms of accuracy and speed in generating prime numbers on a large scale. The results of this study indicate The Sieve of Eratosthenes, Atkins and Sundaram algorithms can generate large numbers of prime numbers with good accuracy, this was tested by the Fermat Primality Test Algorithm. The conclusion that can be drawn from this study, The Sieve of Eratosthenes have a faster time to generate prime numbers on a large scale than the other two algorithms.

Integer Factorization – Cryptology Meets Number Theory

Integer factorization is one of the oldest mathematical problems. Initially, the interest in factorization was motivated by curiosity about be­haviour of prime numbers, which are the basic building blocks of all other integers. Early factorization algorithms were not very efficient. However, this dramatically has changed after the invention of the well-known RSA public-key cryptosystem. The reason for this was simple. Finding an efficient fac­toring algorithm is equivalent to breaking RSA. The work overviews development of integer factoring algorithms. It starts from the classical sieve of Eratosthenes, covers the Fermat algorithm and explains the quadratic sieve, which is a good representative of modern fac­toring algorithms. The progress in factoring is illustrated by examples of RSA challenge moduli, which have been factorized by groups of mathemati­cians and cryptographers. Shor's quantum factorization algorithm with poly­nomial complexity is described and the impact on public-key encryption is discussed.

Simple Wave-Optical Superpositions as Prime Number Sieves.

We encode the sequence of prime numbers into simple superpositions of identical waves, mimicking the archetypal prime number sieve of Eratosthenes. The primes are identified as zeros accompanied by phase singularities in a physically generated wave field for integer valued momenta. Similarly, primes are encoded in the diffraction pattern from a simple single aperture and in the harmonics of a single vibrating resonator. Further, diffraction physics connections to number theory reveal how to encode all Gaussian primes, twin primes, and how to construct wave fields with amplitudes equal to the divisor function at integer spatial frequencies. Remarkably, all of these basic diffraction phenomena reveal that the naturally irregular sequence of primes can arise from trivially ordered wave superpositions.

Penerapan Kriptografi RSA Dalam Mengamankan File Teks Berbasis PHP

Abstract - Along with the development of communication technology human needs in the use of technology are increasing, especially in data storage. One way to improve security for data is by using cryptographic methods. RSA Algorithm (Rivest Shamir Adleman) is one method in the branch of cryptography, where RSA is a type of asymmetric cryptography that uses 2 keys, namely public and private keys. The problem of increasing the security of the public key and private key in RSA (Rivest Shamir Adlema) is that the Linear Congruential Generator (LCG) method is needed, LCG is used to generate a set of random numbers to n, where a set of random numbers will be taken that have a number value prime. One method for generating prime numbers is The Sieve Of Eratosthenes algorithm, The Sieve Of Eratosthenes algorithm is a classic algorithm for determining all prime numbers until the n-number is specified. The way the The Sieve Of Eratosthenes method works is to eliminate numbers that are not prime numbers, resulting in a collection of prime numbers. The prime number generated by the The Sieve Of Eratosthenes algorithm will be used for the public key and private key in the RSA criterion. Keywords - RSA, LCG, The Sieve of Eratosthenes.

Tecnologias digitais da informação e comunicação e história da matemática: uma abordagem por meio do Crivo de Eratóstenes

Muitos docentes não se sentem seguros ou acreditam que não têm tempo o suficiente para o desenvolvimento de atividades utilizando metodologias de ensino diferentes. Assim, é necessário que ações de formação sejam implementadas com o intuito de oferecer oportunidades de vivenciar abordagens diferenciadas para o ensino. Essa pesquisa se insere no âmbito dessa questão. Os autores desenvolveram um objeto de aprendizagem utilizando a ferramenta histórica conhecida como Crivo de Eratóstenes, visando também incluir a história da matemática como metodologia de ensino. O objeto desenvolvido foi aplicado com uma turma de alunos da licenciatura em matemática e professores da rede básica de ensino do Estado do Paraná durante a realização de um curso de extensão numa universidade pública do norte do Paraná. O curso tinha como objetivo tratar sobre o uso da lousa digital interativa na educação básica. Os cursistas tiveram a oportunidade de criar seus próprios objetos de aprendizagem para aplicação na lousa digital interativa e também operaram o objeto que foi desenvolvido pelos autores para esta pesquisa. No final, os cursistas responderam a um questionário e as suas respostas foram analisadas e discutidas. Os autores concluíram que é importante para o professor a busca por atualizações por meio de cursos de formação continuada e procurar suporte para o desenvolvimento de atividades que utilizem essas novas tendências, que estão presentes na sala de aula. Concluíram também que o professor é capaz de desenvolver um objeto de aprendizagem possível de ser usado pedagogicamente em suas aulas.