Two Compact Incremental Prime Sieves

TL;DR

This paper introduces two new prime sieves: a practical incremental sieve with efficient time and space complexity, and a modified Atkin-Bernstein sieve that is sublinear, compact, and incremental, solving a long-standing open problem.

Contribution

It presents the rolling sieve, a practical incremental prime sieve, and modifies the Atkin-Bernstein sieve to be sublinear, compact, and incremental, addressing previous open challenges.

Findings

Rolling sieve operates in O(n log log n) time with O(√n) space.

Modified Atkin-Bernstein sieve is sublinear, compact, and incremental.

Both sieves improve efficiency and practicality for prime computation.

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.