In Files

  • prime.rb

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? .

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 prime.rb, line 135
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 prime.rb, line 171
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 prime.rb, line 147
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 prime.rb, line 201
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