On the complexity of computing prime tables

Martín Farach-Colton, Meng Tsung Tsai

    Research output: Chapter in Book/Report/Conference proceedingConference contribution


    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.

    Original languageEnglish (US)
    Title of host publicationAlgorithms and Computation - 26th International Symposium, ISAAC 2015, Proceedings
    EditorsKhaled Elbassioni, Kazuhisa Makino
    PublisherSpringer Verlag
    Number of pages12
    ISBN (Print)9783662489703
    StatePublished - 2015
    Event26th International Symposium on Algorithms and Computation, ISAAC 2015 - Nagoya, Japan
    Duration: Dec 9 2015Dec 11 2015

    Publication series

    NameLecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
    ISSN (Print)0302-9743
    ISSN (Electronic)1611-3349


    Conference26th International Symposium on Algorithms and Computation, ISAAC 2015


    • Factorial
    • Lower bound
    • Multiplication
    • Prime tables

    ASJC Scopus subject areas

    • Theoretical Computer Science
    • General Computer Science


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