NAME
Math::Prime::XS - Detect and calculate prime numbers with deterministic
tests
SYNOPSIS
use Math::Prime::XS ':all';
# or
use Math::Prime::XS qw(is_prime primes mod_primes sieve_primes sum_primes trial_primes);
print "prime" if is_prime(59);
@all_primes = primes(100);
@range_primes = primes(30, 70);
@all_primes = mod_primes(100);
@range_primes = mod_primes(30, 70);
@all_primes = sieve_primes(100);
@range_primes = sieve_primes(30, 70);
@all_primes = sum_primes(100);
@range_primes = sum_primes(30, 70);
@all_primes = trial_primes(100);
@range_primes = trial_primes(30, 70);
DESCRIPTION
"Math::Prime::XS" detects and calculates prime numbers by either
applying Modulo operator division, the Sieve of Eratosthenes, a
Summation calculation or Trial division.
FUNCTIONS
is_prime
is_prime($number);
Returns true if the number is prime, false if not.
The XS function invoked within "is_prime()" is subject to change
(currently it's an all-XS trial division skipping multiples of 2,3,5).
primes
@all_primes = primes($number);
@range_primes = primes($base, $number);
Returns all primes for the given number or primes between the base and
number.
The resolved function called is subject to change (currently
"sieve_primes()").
count_primes
$count = count_primes($number);
$count = count_primes($base, $number);
Return a count of primes from 0 to $number, or from $base to $number,
inclusive. The arguments are the same as "primes()" but the return is
just a count of the primes.
SPECIFIC ALGORITHMS
mod_primes
@all_primes = mod_primes($number);
@range_primes = mod_primes($base, $number);
Applies the Modulo operator division algorithm:
Divide the number by 2 and all odd numbers <= sqrt(n); if any divides
exactly then the number is not prime.
Returns all primes between 2 and $number, or between $base and $number
(inclusive).
(This function differs from "trial_primes" in that the latter takes some
trouble to divide only by primes below sqrt(n), whereas "mod_primes"
divides by all integers not easily identifiable as composite.)
sieve_primes
@all_primes = sieve_primes($number);
@range_primes = sieve_primes($base, $number);
Applies the Sieve of Eratosthenes algorithm:
One of the most efficient ways to find all the small primes (say, all
those less than 10,000,000) is by using the Sieve of Eratosthenes (ca
240 BC). Make a list of all numbers less than or equal to n (and greater
than one) and strike out the multiples of all primes less than or equal
to the square root of n: the numbers that are left are primes.
Returns all primes for the given number or primes between the base and
number.
sum_primes
@all_primes = sum_primes($number);
@range_primes = sum_primes($base, $number);
Applies the Summation calculation algorithm:
The summation calculation algorithm resembles the modulo operator
division algorithm, but also shares some common properties with the
Sieve of Eratosthenes. For each saved prime smaller than or equal to the
square root of the number, recall the corresponding sum (if none, start
with zero); add the prime to the sum being calculated while the
summation is smaller than the number. If none of the sums equals the
number, then the number is prime.
Returns all primes for the given number or primes between the base and
number.
trial_primes
@all_primes = trial_primes($number);
@range_primes = trial_primes($base, $number);
Applies the Trial division algorithm:
To see if an individual small number is prime, trial division works
well: just divide by all the primes less than or equal to its square
root. For example, to assert 211 is prime, divide by 2, 3, 5, 7, 11 and
13. Since none of these primes divides the number evenly, it is prime.
Returns all primes for the given number or primes between the base and
number.
BENCHMARK
Following output resulted from a benchmark measuring the time to
calculate primes up to 1,000,000 with 100 iterations for each function.
The tests were conducted by the "cmpthese" function of the Benchmark
module.
Rate mod_primes trial_primes sum_primes sieve_primes
mod_primes 1.32/s -- -58% -79% -97%
trial_primes 3.13/s 137% -- -49% -93%
sum_primes 6.17/s 366% 97% -- -86%
sieve_primes 43.3/s 3173% 1284% 602% --
The "Rate" column is the speed in how many times per second, so
"sieve_primes()" is the fastest for this particular test.
EXPORT
Functions
"is_prime(), primes(), mod_primes(), sieve_primes(), sum_primes(),
trial_primes()" are exportable.
Tags
":all - *()"
BUGS & CAVEATS
Note that the order of execution speed for functions may differ from the
benchmarked results when numbers get larger or smaller.
SEE ALSO
,
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