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Two compact incremental prime sieves

Part of:Elementary number theoryComputational number theoryTheory of computing

Published online by Cambridge University Press:  01 November 2015

Jonathan P. Sorenson
Jonathan P. Sorenson*
Affiliation:
Computer Science and Software Engineering, Butler University, Indianapolis, IN 46208, USA email sorenson@butler.edu

Abstract

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A prime sieve is an algorithm that finds the primes up to a bound $n$ . We say that a prime sieve is incremental, if it can quickly determine if $n+1$ is prime after having found all primes up to  $n$ . We say a sieve is compact if it uses roughly $\sqrt{n}$ space or less. In this paper, we present two new results.

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We describe the rolling sieve, a practical, incremental prime sieve that takes $O(n\log \log n)$ time and $O(\sqrt{n}\log n)$ bits of space.

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We also show how to modify the sieve of Atkin and Bernstein from 2004 to obtain a sieve that is simultaneously sublinear, compact, and incremental.

The second result solves an open problem given by Paul Pritchard in 1994.

MSC classification

Primary:11Y16: Algorithms; complexity68Q25: Analysis of algorithms and problem complexity
Secondary:11Y11: Primality11A51: Factorization; primality

Information

Type
Research Article
Information
LMS Journal of Computation and Mathematics , Volume 18 , Issue 1 , 2015, pp. 675 - 683
Copyright
© The Author 2015 

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