On the complexity of computing prime tables

TL;DR

This paper presents two new algorithms for computing prime tables with improved complexity bounds, significantly enhancing the efficiency of related large arithmetic computations like factorials and binomial coefficients.

Contribution

It introduces two algorithms for prime table computation with complexities of O(M(n log n)) and O(n log^2 n / log log n), improving previous methods by speeding up Atkin's sieve.

Findings

The second algorithm outperforms previous algorithms by a factor of log^2 log n.

Fast prime tables accelerate calculations of n! and binomial coefficients.

Lower bounds on factorial computation complexity are established.

Abstract

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 $\binom{n}{n/2}$ 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 \cite{Furer07,De08}. 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 $n\log n 2^{O(\log^*n)}$, 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 $\binom{n}{n/2}$. Finally, we show that computing the factorial takes $\Omega(M(n \log^{4/7 - \epsilon} n))$ for any constant $\epsilon > 0$ assuming only multiplication is allowed.