# NAME

Math::Primality - Advanced Primality Algorithms using GMP

# VERSION

Version 0.04

# SYNOPSIS

```
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);
```

# DESCRIPTION

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.

# EXPORT

# FUNCTIONS

## is_pseudoprime($n,$b)

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
}
```

### Details

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

## is_strong_pseudoprime($n,$b)

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
}
```

### Details

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

### Notes

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

## is_strong_lucas_pseudoprime($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
}
```

### Details

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

### Notes

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

## is_prime($n)

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
}
```

### Details

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.

### Notes

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.

## next_prime($n)

Given a number, produces the next prime number.

` my $q = next_prime($n);`

### Details

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

### Notes

Checking of primality is implemented by is_prime()

## prev_prime($n)

Given a number, produces the previous prime number.

` my $q = prev_prime($n);`

### Details

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

### Notes

Checking of primality is implemented by is_prime()

## prime_count($n)

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
```

### Details

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.

### Notes

Checking of primality is implemented by is_prime()

# AUTHORS

Jonathan Leto, `<jonathan at leto.net>`

Bob Kuo, `<bobjkuo at gmail.com>`

# BUGS

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.

# THANKS

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.

# SUPPORT

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

RT: CPAN's request tracker

AnnoCPAN: Annotated CPAN documentation

CPAN Ratings

Search CPAN

# ACKNOWLEDGEMENTS

# COPYRIGHT & LICENSE

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.