145 research outputs found

    Approximately counting semismooth integers

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    An integer nn is (y,z)(y,z)-semismooth if n=pmn=pm where mm is an integer with all prime divisors ≤y\le y and pp is 1 or a prime ≤z\le z. arge quantities of semismooth integers are utilized in modern integer factoring algorithms, such as the number field sieve, that incorporate the so-called large prime variant. Thus, it is useful for factoring practitioners to be able to estimate the value of Ψ(x,y,z)\Psi(x,y,z), the number of (y,z)(y,z)-semismooth integers up to xx, so that they can better set algorithm parameters and minimize running times, which could be weeks or months on a cluster supercomputer. In this paper, we explore several algorithms to approximate Ψ(x,y,z)\Psi(x,y,z) using a generalization of Buchstab's identity with numeric integration.Comment: To appear in ISSAC 2013, Boston M

    The I/O Complexity of Computing Prime Tables

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    International audienceWe revisit classical sieves for computing primes and analyze their performance in the external-memory model. Most prior sieves are analyzed in the RAM model, where the focus is on minimizing both the total number of operations and the size of the working set. The hope is that if the working set fits in RAM, then the sieve will have good I/O performance, though such an outcome is by no means guaranteed by a small working-set size. We analyze our algorithms directly in terms of I/Os and operations. In the external-memory model, permutation can be the most expensive aspect of sieving, in contrast to the RAM model, where permutations are trivial. We show how to implement classical sieves so that they have both good I/O performance and good RAM performance, even when the problem size N becomes huge—even superpolynomially larger than RAM. Towards this goal, we give two I/O-efficient priority queues that are optimized for the operations incurred by these sieves

    On Generating Prime Numbers Efficiently

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    The prime numbers can be considered as the building blocks of natural numbers, having innumerable applications in number theory and cryptography. There exist multiple different sieving algorithms for the generation of prime numbers. In this thesis, an elementary modular result is utilized to construct an analytically useful generator function and its inverse function. The functions are used to generate a (log)log-linear time complexity prime sieving algorithm which is further optimized to be of linear time complexity. The constructed algorithms and their operation are studied and the linear implementations in JS, Python and C++ are compared to other prime sieves.Alkulukuja voidaan pitää luonnollisten lukujen rakennuspalikoina joilla on lukemattomia sovelluksia lukuteoriassa ja kryptografiassa. Alkulukujen luomiseen on olemassa useita erilaisia seulonta-algoritmeja. Tässä opinnäytetyössä käytetään modulaarista perustulosta analyyttisesti hyödyllisten kehitysfunktion ja sen käänteisfunktion luomiseen. Funktioiden avulla luodaan aikakompleksisuudeltaan (log)log-lineaarinen alkulukuseula, joka optimoidaan lineaariseksi. Rakennettuja algoritmeja ja niiden toimintaa tarkastellaan ja lineaarista implementaatiota JS, Python ja C++ ohjelmointikielillä verrataan toisiin alkulukuseuloihin

    What is the best approach to counting primes?

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    As long as people have studied mathematics, they have wanted to know how many primes there are. Getting precise answers is a notoriously difficult problem, and the first suitable technique, due to Riemann, inspired an enormous amount of great mathematics, the techniques and insights permeating many different fields. In this article we will review some of the best techniques for counting primes, centering our discussion around Riemann's seminal paper. We will go on to discuss its limitations, and then recent efforts to replace Riemann's theory with one that is significantly simpler.Comment: To appear in a volume dedicated to the MAA Centennial in 201

    Computing π(N)\pi(N): An elementary approach in O~(N)\tilde{O}(\sqrt{N}) time

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    We present an efficient and elementary algorithm for computing the number of primes up to NN in O~(N)\tilde{O}(\sqrt N) time, improving upon the existing combinatorial methods that require O~(N2/3)\tilde{O}(N ^ {2/3}) time. Our method has a similar time complexity to the analytical approach to prime counting, while avoiding complex analysis and the use of arbitrary precision complex numbers. While the most time-efficient version of our algorithm requires O~(N)\tilde{O}(\sqrt N) space, we present a continuous space-time trade-off, showing, e.g., how to reduce the space complexity to O~(N3)\tilde{O}(\sqrt[3]{N}) while slightly increasing the time complexity to O~(N8/15)\tilde{O}(N^{8/15}). We apply our techniques to improve the state-of-the-art complexity of elementary algorithms for computing other number-theoretic functions, such as the the Mertens function (in O~(N)\tilde{O}(\sqrt N) time compared to the known O~(N0.6)\tilde{O}(N^{0.6})), summing Euler's totient function, counting square-free numbers and summing primes. Implementation code is provided

    Space station operating system study

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    The current phase of the Space Station Operating System study is based on the analysis, evaluation, and comparison of the operating systems implemented on the computer systems and workstations in the software development laboratory. Primary emphasis has been placed on the DEC MicroVMS operating system as implemented on the MicroVax II computer, with comparative analysis of the SUN UNIX system on the SUN 3/260 workstation computer, and to a limited extent, the IBM PC/AT microcomputer running PC-DOS. Some benchmark development and testing was also done for the Motorola MC68010 (VM03 system) before the system was taken from the laboratory. These systems were studied with the objective of determining their capability to support Space Station software development requirements, specifically for multi-tasking and real-time applications. The methodology utilized consisted of development, execution, and analysis of benchmark programs and test software, and the experimentation and analysis of specific features of the system or compilers in the study

    Pattern and Mathematics: Math Enrichment Activities for Gifted Fourth, Fifth and Sixth Grade Children

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    The purpose of this project is to provide assistance to the elementary math teacher in meeting the needs of the student gifted in the area of math. A collection of activities is provided to use with gifted intermediate students and should serve as an example of the type of activities appropriate for the gifted student. These activities would be most appropriately used with gifted students in grades four through six. The activities would be of greatest benefit if students were grouped in a homogeneous manner with one teacher taking responsibility for their math needs, Homogeneous grouping would allow for faster-paced instruction and stimulating interaction among the students as well as relieving other teachers of this responsibility and providing continuity to the program. The activities encourage use of higher level thinking and exploration with a variety of materials. The student is encouraged to think and work as a mathematician does, dealing with both inductive and deductive reasoning
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