Faster enumeration of primes (opens in new tab)

We describe several new algorithms for finding all prime numbers up to a given bound $N$, achieving the first ever speedup by a positive power of $\log N$ over the ancient sieve of Eratosthenes. The fastest version, which is not fully rigorous, runs in \[ N (\log \log N)^{1+o(1)} \] bit operations when analysed in the multitape Turing model. This improves on the best existing algorithms due to Pritchard (1981), Atkin--Bernstein (2004) and Serg...