Reducing the space used by the sieve of Eratosthenes when factoring

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