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 (nn/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 log2 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 log2 log n. Our fast prime-table algorithms speed up both the computation of n! and (n/n2). Finally, we show that computing the factorial takes Ω(M(n log4/7−εn)) for any constant ε > 0 assuming only multiplication is allowed.