Reducing the Space Used by the Sieve of Eratosthenes When Factoring

TL;DR

This paper introduces a space-efficient version of the sieve of Eratosthenes that can factor all integers up to x within near-linear time and significantly reduced space, with practical performance validation.

Contribution

It presents a novel space-optimized sieve of Eratosthenes algorithm capable of factoring integers up to x within specified time and space bounds, improving previous methods.

Findings

Achieves $O(x \, \log\log x)$ time complexity.

Uses at most $O(\sqrt{x}/\log\log x)$ bits of space.

Performs well in practical implementations.

Abstract

We present a version of the sieve of Eratosthenes that can factor all integers $\le x$ in $O(x \log\log x)$ arithmetic operations using at most $O(\sqrt{x}/\log\log x)$ bits of space. This is an improved space bound under the condition that the algorithm takes at most $O(x\log\log x)$ time. We also show our algorithm performs well in practice.