# 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.

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

## 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.

http://www.geraldbuehler.de/primzahlen/

## 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.

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

# 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

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

# AUTHOR

Steven Schubiger <schubiger@cpan.org>

# LICENSE

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