Two Compact Incremental Prime Sieves

Jonathan P. Sorenson

doi:10.1112/s1461157015000194

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arXiv · 1503.02592

Two Compact Incremental Prime Sieves

Jonathan P. Sorenson

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Abstract

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: (1) 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, and (2) We show how to modify the sieve of Atkin and Bernstein (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.

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Jonathan P. Sorenson. 2015-03-09. Two Compact Incremental Prime Sieves. https://doi.org/10.1112/s1461157015000194

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