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+#
+# = prime.rb
+#
+# Prime numbers and factorization library.
+#
+# Copyright::
+#   Copyright (c) 1998-2008 Keiju ISHITSUKA(SHL Japan Inc.)
+#   Copyright (c) 2008 Yuki Sonoda (Yugui) <yugui@yugui.jp>
+#
+# Documentation::
+#   Yuki Sonoda
+#
+
+require "singleton"
+require "forwardable"
+
+class Integer
+  # Re-composes a prime factorization and returns the product.
+  #
+  # See Prime#int_from_prime_division for more details.
+  def Integer.from_prime_division(pd)
+    Prime.int_from_prime_division(pd)
+  end
+
+  # Returns the factorization of +self+.
+  #
+  # See Prime#prime_division for more details.
+  def prime_division(generator = Prime::Generator23.new)
+    Prime.prime_division(self, generator)
+  end
+
+  # Returns true if +self+ is a prime number, else returns false.
+  def prime?
+    Prime.prime?(self)
+  end
+
+  # Iterates the given block over all prime numbers.
+  #
+  # See +Prime+#each for more details.
+  def Integer.each_prime(ubound, &block) # :yields: prime
+    Prime.each(ubound, &block)
+  end
+end
+
+#
+# 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+# +Prime+.new is obsolete.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 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? .
+
+class Prime
+  include Enumerable
+  @the_instance = Prime.new
+
+  # obsolete. Use +Prime+::+instance+ or class methods of +Prime+.
+  def initialize
+    @generator = EratosthenesGenerator.new
+    extend OldCompatibility
+    warn "Prime::new is obsolete. use Prime::instance or class methods of Prime."
+  end
+
+  class << self
+    extend Forwardable
+    include Enumerable
+    # Returns the default instance of Prime.
+    def instance; @the_instance end
+
+    def method_added(method) # :nodoc:
+      (class<< self;self;end).def_delegator :instance, method
+    end
+  end
+
+  # 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+.
+  #
+  # == Note
+  #
+  # +Prime+.+new+ returns an object extended by +Prime+::+OldCompatibility+
+  # in order to be compatible with 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+.
+  def each(ubound = nil, generator = EratosthenesGenerator.new, &block)
+    generator.upper_bound = ubound
+    generator.each(&block)
+  end
+
+
+  # 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.
+  def prime?(value, generator = Prime::Generator23.new)
+    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
+
+  # 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 <tt>[[p_1, e_1], [p_2, e_2], ...., [p_n, e_n]]</tt>, it returns:
+  #
+  #   p_1**e_1 * p_2**e_2 * .... * p_n**e_n.
+  #
+  #   Prime.int_from_prime_division([[2,2], [3,1]])  #=> 12
+  def int_from_prime_division(pd)
+    pd.inject(1){|value, (prime, index)|
+      value * prime**index
+    }
+  end
+
+  # 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]]
+  #
+  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
+
+  # An abstract class for enumerating pseudo-prime numbers.
+  #
+  # Concrete subclasses should override succ, next, rewind.
+  class PseudoPrimeGenerator
+    include Enumerable
+
+    def initialize(ubound = nil)
+      @ubound = ubound
+    end
+
+    def upper_bound=(ubound)
+      @ubound = ubound
+    end
+    def upper_bound
+      @ubound
+    end
+
+    # returns the next pseudo-prime number, and move the internal
+    # position forward.
+    #
+    # +PseudoPrimeGenerator+#succ raises +NotImplementedError+.
+    def succ
+      raise NotImplementedError, "need to define `succ'"
+    end
+
+    # alias of +succ+.
+    def next
+      raise NotImplementedError, "need to define `next'"
+    end
+
+    # Rewinds the internal position for enumeration.
+    #
+    # See +Enumerator+#rewind.
+    def rewind
+      raise NotImplementedError, "need to define `rewind'"
+    end
+
+    # Iterates the given block for each prime number.
+    def each
+      return self.dup unless block_given?
+      if @ubound
+        last_value = nil
+        loop do
+          prime = succ
+          break last_value if prime > @ubound
+          last_value = yield prime
+        end
+      else
+        loop do
+          yield succ
+        end
+      end
+    end
+
+    # see +Enumerator+#with_index.
+    alias with_index each_with_index
+
+    # see +Enumerator+#with_object.
+    def with_object(obj)
+      return enum_for(:with_object) unless block_given?
+      each do |prime|
+        yield prime, obj
+      end
+    end
+  end
+
+  # An implementation of +PseudoPrimeGenerator+.
+  #
+  # Uses +EratosthenesSieve+.
+  class EratosthenesGenerator < PseudoPrimeGenerator
+    def initialize
+      @last_prime_index = -1
+      super
+    end
+
+    def succ
+      @last_prime_index += 1
+      EratosthenesSieve.instance.get_nth_prime(@last_prime_index)
+    end
+    def rewind
+      initialize
+    end
+    alias next succ
+  end
+
+  # An implementation of +PseudoPrimeGenerator+ which uses
+  # a prime table generated by trial division.
+  class TrialDivisionGenerator<PseudoPrimeGenerator
+    def initialize
+      @index = -1
+      super
+    end
+
+    def succ
+      TrialDivision.instance[@index += 1]
+    end
+    def rewind
+      initialize
+    end
+    alias next succ
+  end
+
+  # Generates all integers which are greater than 2 and
+  # are not divisible by either 2 or 3.
+  #
+  # This is a pseudo-prime generator, suitable on
+  # checking primality of an integer by brute force
+  # method.
+  class Generator23<PseudoPrimeGenerator
+    def initialize
+      @prime = 1
+      @step = nil
+      super
+    end
+
+    def succ
+      if (@step)
+        @prime += @step
+        @step = 6 - @step
+      else
+        case @prime
+        when 1; @prime = 2
+        when 2; @prime = 3
+        when 3; @prime = 5; @step = 2
+        end
+      end
+      return @prime
+    end
+    alias next succ
+    def rewind
+      initialize
+    end
+  end
+
+  # Internal use. An implementation of prime table by trial division method.
+  class TrialDivision
+    include Singleton
+
+    def initialize # :nodoc:
+      # These are included as class variables to cache them for later uses.  If memory
+      #   usage is a problem, they can be put in Prime#initialize as instance variables.
+
+      # There must be no primes between @primes[-1] and @next_to_check.
+      @primes = [2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101]
+      # @next_to_check % 6 must be 1.
+      @next_to_check = 103            # @primes[-1] - @primes[-1] % 6 + 7
+      @ulticheck_index = 3            # @primes.index(@primes.reverse.find {|n|
+      #   n < Math.sqrt(@@next_to_check) })
+      @ulticheck_next_squared = 121   # @primes[@ulticheck_index + 1] ** 2
+    end
+
+    # Returns the cached prime numbers.
+    def cache
+      return @primes
+    end
+    alias primes cache
+    alias primes_so_far cache
+
+    # Returns the +index+th prime number.
+    #
+    # +index+ is a 0-based index.
+    def [](index)
+      while index >= @primes.length
+        # Only check for prime factors up to the square root of the potential primes,
+        #   but without the performance hit of an actual square root calculation.
+        if @next_to_check + 4 > @ulticheck_next_squared
+          @ulticheck_index += 1
+          @ulticheck_next_squared = @primes.at(@ulticheck_index + 1) ** 2
+        end
+        # Only check numbers congruent to one and five, modulo six. All others
+
+        #   are divisible by two or three.  This also allows us to skip checking against
+        #   two and three.
+        @primes.push @next_to_check if @primes[2..@ulticheck_index].find {|prime| @next_to_check % prime == 0 }.nil?
+        @next_to_check += 4
+        @primes.push @next_to_check if @primes[2..@ulticheck_index].find {|prime| @next_to_check % prime == 0 }.nil?
+        @next_to_check += 2
+      end
+      return @primes[index]
+    end
+  end
+
+  # Internal use. An implementation of eratosthenes' sieve
+  class EratosthenesSieve
+    include Singleton
+
+    def initialize
+      @primes = [2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101]
+      # @max_checked must be an even number
+      @max_checked = @primes.last + 1
+    end
+
+    def get_nth_prime(n)
+      compute_primes while @primes.size <= n
+      @primes[n]
+    end
+
+    private
+    def compute_primes
+      # max_segment_size must be an even number
+      max_segment_size = 1e6.to_i
+      max_cached_prime = @primes.last
+      # do not double count primes if #compute_primes is interrupted
+      # by Timeout.timeout
+      @max_checked = max_cached_prime + 1 if max_cached_prime > @max_checked
+
+      segment_min = @max_checked
+      segment_max = [segment_min + max_segment_size, max_cached_prime * 2].min
+      root = Integer(Math.sqrt(segment_max).floor)
+
+      sieving_primes = @primes[1 .. -1].take_while { |prime| prime <= root }
+      offsets = Array.new(sieving_primes.size) do |i|
+        (-(segment_min + 1 + sieving_primes[i]) / 2) % sieving_primes[i]
+      end
+
+      segment = ((segment_min + 1) .. segment_max).step(2).to_a
+      sieving_primes.each_with_index do |prime, index|
+        composite_index = offsets[index]
+        while composite_index < segment.size do
+          segment[composite_index] = nil
+          composite_index += prime
+        end
+      end
+
+      segment.each do |prime|
+        @primes.push prime unless prime.nil?
+      end
+      @max_checked = segment_max
+    end
+  end
+
+  # Provides a +Prime+ object with compatibility to Ruby 1.8 when instantiated via +Prime+.+new+.
+  module OldCompatibility
+    # Returns the next prime number and forwards internal pointer.
+    def succ
+      @generator.succ
+    end
+    alias next succ
+
+    # Overwrites Prime#each.
+    #
+    # Iterates the given block over all prime numbers. Note that enumeration
+    # starts from the current position of internal pointer, not rewound.
+    def each
+      return @generator.dup unless block_given?
+      loop do
+        yield succ
+      end
+    end
+  end
+end