Two compact incremental prime sieves

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.–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.–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.