Math::Primality - Advanced Primality Algorithms using GMP

Version 0.04

    use Math::Primality qw/:all/;

    my $t1 = is_pseudoprime($x,$base);
    my $t2 = is_strong_pseudoprime($x);

    print "Prime!" if is_prime($outrageously_large_prime);

    my $t3 = next_prime($x); 

Math::Primality implements is_prime() and next_prime() as a replacement for Math::PARI::is_prime(). It uses the GMP library through Math::GMPz. The is_prime() method is actually a Baillie-PSW primality test which consists of two steps:

* Perform a strong Miller-Rabin probable prime test (base 2) on N

* Perform a strong Lucas-Selfridge test on N (using the parameters suggested by Selfridge)

At any point the function may return 2 which means N is definitely composite. If not, N has passed the strong Baillie-PSW test and is either prime or a strong Baillie-PSW pseudoprime. To date no counterexample (Baillie-PSW strong pseudoprime) is known to exist for N < 10^15. Baillie-PSW requires O((log n)^3) bit operations. See http://www.trnicely.net/misc/bpsw.html for a more thorough introduction to the Baillie-PSW test. Also see http://mpqs.free.fr/LucasPseudoprimes.pdf for a more theoretical introduction to the Baillie-PSW test.

Returns true if $n is a base $b pseudoprime, otherwise false. The variable $n should be a Perl integer or Math::GMPz object.

The default base of 2 is used if no base is given. Base 2 pseudoprimes are often called Fermat pseudoprimes.

    if ( is_pseudoprime($n,$b) ) {
        # it's a pseudoprime
    } else {
        # not a psuedoprime
    }

A pseudoprime is a number that satisfies Fermat's Little Theorm, that is, $b^ ($n - 1) = 1 mod $n.

Returns true if $n is a base $b strong pseudoprime, false otherwise. The variable $n should be a Perl integer or a Math::GMPz object. Strong psuedoprimes are often called Miller-Rabin pseudoprimes.

The default base of 2 is used if no base is given.

    if ( is_strong_pseudoprime($n,$b) ) {
        # it's a strong pseudoprime
    } else {
        # not a strong psuedoprime
    }

A strong pseudoprime to $base is an odd number $n with ($n - 1) = $d * 2^$s that either satisfies

* $base^$d = 1 mod $n

* $base^($d * 2^$r) = -1 mod $n, for $r = 0, 1, ..., $s-1

The second condition is checked by sucessive squaring $base^$d and reducing that mod $n.

Returns true if $n is a strong Lucas-Selfridge pseudoprime, false otherwise. The variable $n should be a Perl integer or a Math::GMPz object.

    if ( is_strong_lucas_pseudoprime($n) ) {
        # it's a strong Lucas-Selfridge pseudoprime
    } else {
        # not a strong Lucas-Selfridge psuedoprime
        # i.e. definitely composite
    }

If we let

* $D be the first element of the sequence 5, -7, 9, -11, 13, ... for which ($D/$n) = -1. Let $P = 1 and $Q = (1 - $D) /4

* U($P, $Q) and V($P, $Q) be Lucas sequences

* $n + 1 = $d * 2^$s + 1

Then a strong Lucas-Selfridge pseudoprime is an odd, non-perfect square number $n with that satisfies either

* U_$d = 0 mod $n

* V_($d * 2^$r) = 0 mod $n, for $r = 0, 1, ..., $s-1

($d/$n) refers to the Legendre symbol.

Returns 2 if $n is definitely prime, 1 is $n is a probable prime, 0 if $n is composite.

    if ( is_prime($n) ) {
        # it's a prime
    } else {
        # definitely composite
    }

is_prime() is implemented using the BPSW algorithim which is a combination of two probable-prime algorithims, the strong Miller-Rabin test and the strong Lucas-Selfridge test. While no psuedoprime has been found for N < 10^15, this does not mean there is not a pseudoprime. A possible improvement would be to instead implement the AKS test which runs in quadratic time and is deterministic with no false-positives.

The strong Miller-Rabin test is implemented by is_strong_pseudoprime(). The strong Lucas-Selfridge test is implemented by is_strong_lucas_pseudoprime().

We have implemented some optimizations. We have an array of small primes to check all $n <= 257. According to http://primes.utm.edu/prove/prove2_3.html if $n < 9,080,191 is a both a base-31 and a base-73 strong pseudoprime, then $n is prime. If $n < 4,759,123,141 is a base-2, base-7 and base-61 strong pseudoprime, then $n is prime.

Given a number, produces the next prime number.

    my $q = next_prime($n);

Each next greatest odd number is checked until one is found to be prime

Checking of primality is implemented by is_prime()

Given a number, produces the previous prime number.

    my $q = prev_prime($n);

Each previous odd number is checked until one is found to be prime. prev_prime(2) or for any number less than 2 returns undef

Checking of primality is implemented by is_prime()

Returns the number of primes less than or equal to $n.

    my $count = prime_count(1000);          # $count = 168
    my $bigger_count = prime_count(10000);  # $bigger_count = 1229

This is implemented with a simple for loop. The Meissel, Lehmer, Lagarias, Miller, Odlyzko method is considerably faster. A paper can be found at http://www.ams.org/mcom/1996-65-213/S0025-5718-96-00674-6/S0025-5718-96-00674-6.pdf that describes this method in rigorous detail.

Checking of primality is implemented by is_prime()

Jonathan Leto, <jonathan at leto.net> Bob Kuo, <bobjkuo at gmail.com>

Please report any bugs or feature requests to bug-math-primality at rt.cpan.org, or through the web interface at http://rt.cpan.org/NoAuth/ReportBug.html?Queue=Math::Primality. I will be notified, and then you'll automatically be notified of progress on your bug as I make changes.

The algorithms in this module have been ported from the C source code in bpsw1.zip by Thomas R. Nicely, available at http://www.trnicely.net/misc/bpsw.html or in the spec/bpsw directory of the Math::Primality source code. Without his research this module would not exist.

The Math::GMPz module that interfaces with the GMP C-library was written and is maintained by Sysiphus. Without his work, our work would be impossible.

You can find documentation for this module with the perldoc command.

    perldoc Math::Primality

You can also look for information at:

* Math::Primality on Github

http://github.com/leto/math--primality/tree/master

* RT: CPAN's request tracker

http://rt.cpan.org/NoAuth/Bugs.html?Dist=Math::Primality

* AnnoCPAN: Annotated CPAN documentation

http://annocpan.org/dist/Math::Primality

* CPAN Ratings

http://cpanratings.perl.org/d/Math::Primality

* Search CPAN

http://search.cpan.org/dist/Math::Primality

Copyright 2009 Jonathan Leto, all rights reserved.

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