NAME
    Math::Prime::XS - Calculate/detect prime numbers with deterministic
    tests

SYNOPSIS
     use Math::Prime::XS qw(primes is_prime);
 
     @allprimes  = primes(9);
     @someprimes = primes(4,9);
 
     if (is_prime(11)) { # do something }

DESCRIPTION
    Math::Prime::XS calculates/detects prime numbers by either applying
    Modulo operator division, the Sieve of Eratosthenes, Trial division or a
    Summing calculation.

FUNCTIONS
  primes
    Takes an integer and calculates the primes from 0 <= integer. Optionally
    an integer may be provided as first argument which will function as
    limit. Calculation then will take place within the range of the limit
    and the integer. Calls "sum_primes()" beneath the surface.

  is_prime
    Takes an integer as input and returns 1 if integer is prime, undef if it
    isn't. The underlying algorithm has been taken from "sum_primes()".

  mod_primes
    Applies the Modulo operator division and provides same functionality and
    interface as "primes()". Divides the number by all n less or equal then
    the number; if the number gets exactly two times divided by rest null,
    then the number is prime, otherwise not.

  sieve_primes
    Applies the Sieve of Erathosthenes and provides same functionality and
    interface as "primes()". The most efficient way to find all of the small
    primes (say all those less than 10,000,000) is by using the Sieve of
    Eratosthenes (ca 240 BC): Make a list of all the integers less than or
    equal to n (and greater than one). Strike out the multiples of all
    primes less than or equal to the square root of n, then the numbers that
    are left are the primes.

    <http://primes.utm.edu/glossary/page.php?sort=SieveOfEratosthenes>

  sum_primes
    Applies a Summing calculation that is somehow similar to
    "trial_primes()"; provides same functionality and interface as
    "primes()". Compared to "trial_primes()", Trial division is being
    omitted and replaced by an addition of primes less than the number's
    square root. If one of the "multiples" equals the number, then the
    number is not prime, otherwise, it is. This algorithm is a somewhat
    hybrid between the Sieve of Eratosthenes and Trial division.

    <http://www.geraldbuehler.de/primzahlen>

  trial_primes
    Applies Trial division and provides the same functionality and interface
    as "primes()". To see if an individual small integer is prime, Trial
    division works well: just divide by all the primes less than (or equal
    to) its square root. For example, to show 211 is prime, just divide by
    2, 3, 5, 7, 11, and 13. Since none of these divides the number evenly,
    it is a prime.

    <http://primes.utm.edu/glossary/page.php?sort=TrialDivision>

BENCHMARK
    If one appends "_primes" to the names on the left, one gets the full
    subnames. Following benchmark output refers to output generated by the
    "cmpthese()" function of the Benchmark module.

    Calculation results:

    primes <= 4000, one iteration:

              Rate sieve   mod trial   sum
     sieve 0.333/s    --  -97%  -98%  -99%
     mod    11.9/s 3478%    --  -33%  -57%
     trial  17.9/s 5277%   50%    --  -35%
     sum    27.6/s 8186%  132%   54%    --

    primes <= 8000, one iteration:

                 Rate sieve   mod   sum trial
     sieve 7.71e-02/s    --  -98%  -99%  -99%
     mod       3.31/s 4188%    --  -53%  -54%
     sum       7.00/s 8979%  112%    --   -2%
     trial     7.14/s 9164%  116%    2%    --

    Bear in mind, that these results are not too reliable as the author
    could neither increase the number nor the iteration count provided,
    because if he attempted to do so, perl would report "Out of memory!",
    which was most likely caused by the Sieve of Eratosthenes algorithm,
    which is rather memory exhaustive by implementation. The Sieve of
    Eratosthenes is expected to be the slowest, followed by the Modulo
    operator division, then either Summing calculation or Trial division
    (dependant upon the iterations) followed by its counterpart.

EXPORT
  Functions
    "primes(), is_prime(), mod_primes(), sieve_primes(), sum_primes(),
    trial_primes()" are exportable.

  Tags
    ":all - *()"

SEE ALSO
    <http://primes.utm.edu>,
    <http://www.it.fht-esslingen.de/~schmidt/vorlesungen/kryptologie/seminar
    /ws9798/html/prim/prim-1.html>

AUTHOR
    Steven Schubiger <schubiger@cpan.org>

LICENSE
    This program is free software; you may redistribute it and/or modify it
    under the same terms as Perl itself.

    See <http://www.perl.com/perl/misc/Artistic.html>

