# Prime

Class

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

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 an external iterator of prime enumeration which is compatible with an `Enumerator`.

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

`Prime`::`EratosthenesGenerator`

Uses eratosthenes' sieve.

`Prime`::`TrialDivisionGenerator`

Uses the trial division method.

`Prime`::`Generator23`

Generates all positive integers which are not divisible by either 2 or 3. This sequence is very bad as a pseudo-prime sequence. But this is faster and uses much less memory than the other generators. So, it is suitable for factorizing an integer which is not large but has many prime factors. e.g. for `Prime#prime?` .

#### each

#

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 it yields all prime numbers p <= `ubound`.

#### int_from_prime_division

#

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```

#### prime?

#

Returns true if `value` is a prime number, else returns false.

## Parameters

`value`

an arbitrary integer to be checked.

`generator`

optional. A pseudo-prime generator.

#### prime_division

#

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 ascending order. It must generate all prime numbers, but may also generate non prime numbers too.

### 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]]```