The set of all prime numbers.

## Example

``````Prime.each(100) do |prime|
p prime  #=> 2, 3, 5, 7, 11, ...., 97
end
``````

Prime is Enumerable:

``````Prime.first 5 # => [2, 3, 5, 7, 11]
``````

## Retrieving the instance

`Prime`.new is obsolete. Now `Prime` has the default instance and you can access it as `Prime`.instance.

For convenience, each instance method of `Prime`.instance can be accessed as a class method of `Prime`.

e.g.

``````Prime.instance.prime?(2)  #=> true
Prime.prime?(2)           #=> true
``````

## Generators

A “generator” provides an implementation of enumerating pseudo-prime numbers and it remembers the position of enumeration and upper bound. Furthermore, it is a external iterator of prime enumeration which is compatible to an Enumerator.

`Prime`::`PseudoPrimeGenerator` is the base class for generators. There are few implementations of generator.

`Prime`::`EratosthenesGenerator`

Uses eratosthenes's sieve.

`Prime`::`TrialDivisionGenerator`

Uses the trial division method.

`Prime`::`Generator23`

Generates all positive integers which is not divided by 2 nor 3. This sequence is very bad as a pseudo-prime sequence. But this is faster and uses much less memory than other generators. So, it is suitable for factorizing an integer which is not large but has many prime factors. e.g. for #prime? .

Namespace
Methods
E
I
N
P
Included Modules
Class Public methods
instance()

Returns the default instance of Prime.

```# File lib/prime.rb, line 106
def instance; @the_instance end```
new()

obsolete. Use `Prime`::`instance` or class methods of `Prime`.

```# File lib/prime.rb, line 96
def initialize
@generator = EratosthenesGenerator.new
extend OldCompatibility
warn "Prime::new is obsolete. use Prime::instance or class methods of Prime."
end```
Instance Public methods
each(ubound = nil, generator = EratosthenesGenerator.new, &block)

Iterates the given block over all prime numbers.

## Parameters

ubound

Optional. An arbitrary positive number. The upper bound of enumeration. The method enumerates prime numbers infinitely if `ubound` is nil.

`generator`

Optional. An implementation of pseudo-prime generator.

## Return value

An evaluated value of the given block at the last time. Or an enumerator which is compatible to an `Enumerator` if no block given.

## Description

Calls `block` once for each prime number, passing the prime as a parameter.

`ubound`

Upper bound of prime numbers. The iterator stops after yields all prime numbers p <= `ubound`.

## Note

`Prime`.`new` returns a object extended by `Prime`::`OldCompatibility` in order to compatibility to Ruby 1.8, and `Prime`#each is overwritten by `Prime`::`OldCompatibility`#`each`.

`Prime`.`new` is now obsolete. Use `Prime`.`instance`.`each` or simply `Prime`.`each`.

```# File lib/prime.rb, line 147
def each(ubound = nil, generator = EratosthenesGenerator.new, &block)
generator.upper_bound = ubound
generator.each(&block)
end```
int_from_prime_division(pd)

Re-composes a prime factorization and returns the product.

## Parameters

pd

Array of pairs of integers. The each internal pair consists of a prime number – a prime factor – and a natural number – an exponent.

## Example

For `[[p_1, e_1], [p_2, e_2], ...., [p_n, e_n]]`, it returns:

``````p_1**e_1 * p_2**e_2 * .... * p_n**e_n.

Prime.int_from_prime_division([[2,2], [3,1]])  #=> 12``````
```# File lib/prime.rb, line 182
def int_from_prime_division(pd)
pd.inject(1){|value, (prime, index)|
value *= prime**index
}
end```
prime?(value, generator = Prime::Generator23.new)

Returns true if `value` is prime, false for a composite.

## Parameters

value

an arbitrary integer to be checked.

`generator`

optional. A pseudo-prime generator.

```# File lib/prime.rb, line 159
def prime?(value, generator = Prime::Generator23.new)
value = -value if value < 0
return false if value < 2
for num in generator
q,r = value.divmod num
return true if q < num
return false if r == 0
end
end```
prime_division(value, generator= Prime::Generator23.new)

Returns the factorization of `value`.

## Parameters

value

An arbitrary integer.

`generator`

Optional. A pseudo-prime generator. `generator`.succ must return the next pseudo-prime number in the ascendent order. It must generate all prime numbers, but may generate non prime numbers.

### Exceptions

`ZeroDivisionError`

when `value` is zero.

## Example

For an arbitrary integer:

``n = p_1**e_1 * p_2**e_2 * .... * p_n**e_n,``

#prime_division(n) returns:

``````[[p_1, e_1], [p_2, e_2], ...., [p_n, e_n]].

Prime.prime_division(12) #=> [[2,2], [3,1]]``````
```# File lib/prime.rb, line 212
def prime_division(value, generator= Prime::Generator23.new)
raise ZeroDivisionError if value == 0
if value < 0
value = -value
pv = [[-1, 1]]
else
pv = []
end
for prime in generator
count = 0
while (value1, mod = value.divmod(prime)
mod) == 0
value = value1
count += 1
end
if count != 0
pv.push [prime, count]
end
break if value1 <= prime
end
if value > 1
pv.push [value, 1]
end
return pv
end```