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
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