Reducing the space used by the sieve of Eratosthenes when factoring

Abstract

We present a version of the sieve of Eratosthenes that can factor all integers ≤x in O(xlog⁡log⁡x) arithmetic operations using at most O(x/log⁡log⁡x) bits of space. Among algorithms that take the optimal O(xlog⁡log⁡x) time, this new space bound is an improvement of a factor proportional to log⁡xlog⁡log⁡x over the implied previous bound of O(xlog⁡x). We also show our algorithm performs well in practice.