On the Complexity of Computing Prime Tables

摘要

Many large arithmetic computations rely on tables of all primes less than n. For example, the fastest algorithms for computing n! takes time O(M(n log n) + P(n)), where M(n) is the time to multiply two n-bit numbers, and P(n) is the time to compute a prime table up to n. The fastest algorithm to compute ({sup}n {down}n/2) also uses a prime table. We show that it takes time O(M(n) + P(n)). In various models, the best bound on P(n) is greater than M(n log n), given advances in the complexity of multiplication [8,13]. In this paper, we give two algorithms to computing prime tables and analyze their complexity on a multitape Turing machine, one of the standard models for analyzing such algorithms. These two algorithms run in time O(M(n log n)) and O(n log~2 n/log log n), respectively. We achieve our results by speeding up Atkin's sieve. Given that the current best bound on M(n) is n log n2~(O(log~* n)), the second algorithm is faster and improves on the previous best algorithm by a factor of log~2 log n. Our fast prime-table algorithms speed up both the computation of n! and ({sup}n {down}n/2). Finally, we show that computing the factorial takes Ω(M(n log~(4/7-ε) n)) for any constant ε > 0 assuming only multiplication is allowed.