# class Prime

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?` .

### Constants

VERSION

### Public Instance Methods

each(ubound = nil, generator = EratosthenesGenerator.new, &block) click to toggle source

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`.

    # File lib/prime.rb, line 139
    def each(ubound = nil, generator = EratosthenesGenerator.new, &block)
      generator.upper_bound = ubound
      generator.each(&block)
    end

int_from_prime_division(pd) click to toggle source

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 175
    def int_from_prime_division(pd)
      pd.inject(1){|value, (prime, index)|
        value * prime**index
      }
    end

prime?(value, generator = Prime::Generator23.new) click to toggle source

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.

    # File lib/prime.rb, line 151
    def prime?(value, generator = Prime::Generator23.new)
      raise ArgumentError, "Expected a prime generator, got #{generator}" unless generator.respond_to? :each
      raise ArgumentError, "Expected an integer, got #{value}" unless value.respond_to?(:integer?) && value.integer?
      return false if value < 2
      generator.each do |num|
        q,r = value.divmod num
        return true if q < num
        return false if r == 0
      end
    end

prime_division(value, generator = Prime::Generator23.new) click to toggle source

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]]

    # File lib/prime.rb, line 205
    def prime_division(value, generator = Prime::Generator23.new)
      raise ZeroDivisionError if value == 0
      if value < 0
        value = -value
        pv = [[-1, 1]]
      else
        pv = []
      end
      generator.each do |prime|
        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
      pv
    end