view release on metacpan or  search on metacpan

lib/Math/Prime/XS.pm  view on Meta::CPAN

(This function differs from C<trial_primes> in that the latter takes some
trouble to divide only by primes below sqrt(n), whereas C<mod_primes>
divides by all integers not easily identifiable as composite.)

=head2 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.

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

=head2 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.

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

=head2 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.

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

=head1 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 C<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
C<sieve_primes()> is the fastest for this particular test.

=head1 EXPORT

=head2 Functions

C<is_prime(), primes(), mod_primes(), sieve_primes(), sum_primes(), trial_primes()>
are exportable.

=head2 Tags

C<:all - *()>

=head1 BUGS & CAVEATS

Note that the order of execution speed for functions may differ from the
benchmarked results when numbers get larger or smaller.

=head1 SEE ALSO

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

=head1 AUTHOR

Steven Schubiger <schubiger@cpan.org>

=head1 LICENSE

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

See L<http://dev.perl.org/licenses/>

=cut