* lib/mathn.rb (Integer): moved into prime.rb.

  (Prime): ditto.

* lib/prime.rb (Integer): moved from mathn.rb.
  (Integer.each_prime): added.
  (Integer#prime?): added.
  (Prime): moved from mathn.rb.
    Its implmentation was rewritten. see [ruby-dev:35863].
    And patched by Keiju ISHITSUKA <keiju@ishitsuka.com>,
    see [ruby-dev:36128].
  (Prime.new):                     obsolete.
  (Prime.instance):                added.
  (Prime.each):                    added.
  (Prime.int_from_prime_division): added.
  (Prime.prime_division):          added.
  (Prime.prime?):                  added.
    Patch by TOYOFUKU Chikanobu
    <nobu_toyofuku at nifty.com> in [ruby-dev:36067].
  (Prime.cache):                   removed.
  (Prime.primes):                  removed.
  (Prime.primes_so_far):           removed.
  (Prime#int_from_prime_division): added.
  (Prime#prime_division):          added.
  (Prime#prime?):                  added.
  (Prime#primes):                  removed.
  (Prime#primes_so_far):           removed.
  (Prime::PseudoPrmeGenerator):    added.
  (Prime::EratosthenesGenerator):  added.
  (Prime::TrialDivisionGenerator): added.
  (Prime::Generator23):            added.
  (Prime::TrialDivision):          added.
    Extracted from the previous implementation of Prime
    by Keiju ISHITSUKA.
  (Prime::EratosthenesSieve):      added.

* lib/.document (prime.rb): added

* lib/README (prime.rb): added

* test/test_prime.rb: added.

git-svn-id: svn+ssh://ci.ruby-lang.org/ruby/trunk@19095 b2dd03c8-39d4-4d8f-98ff-823fe69b080e

1 files changed, 446 insertions, 0 deletions

diff --git a/lib/prime.rb b/lib/prime.rb
new file mode 100644
index 0000000000..be1c8b42a0
--- /dev/null
+++ b/lib/prime.rb
@@ -0,0 +1,446 @@
+#
+# = 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, false for a composite.
+  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
+#
+# == 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.
+# Futhermore, 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#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
+
+  module OldCompatibility
+    def succ
+      @generator.succ
+    end
+    alias next succ
+
+    def each(&block)
+      loop do
+	yield succ
+      end
+    end
+  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 numer, passing the prime as
+  # a parameter.
+  #
+  # +ubound+::
+  #   Upper bound of prime numbers. The iterator stops after 
+  #   yields all prime numbers p <= +ubound+.
+  def each(ubound = nil, generator = EratosthenesGenerator.new, &block)
+    generator.upper_bound = ubound
+    generator.each(&block)
+  end
+
+
+  # Returns true if +value+ is prime, false for a composite.
+  #
+  # == Parameters
+  # +value+:: an arbitrary integer to be checked.
+  # +generator+:: optional. A pseudo-prime generator.
+  def prime?(value, generator = Prime::Generator23.new)
+    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 [[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
+  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 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]]
+  #
+  def prime_division(value, generator= Prime::Generator23.new)
+    raise ZeroDivisionError if value == 0
+    pv = []
+    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 numbers.
+    # +ubound+:: 
+    def each(&block)
+      return self.dup unless block
+      if @ubound
+	loop do
+	  p = succ
+	  break if p > @ubound
+	  block.call p
+	end
+      else
+	loop do
+	  block.call 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 = nil
+    end
+    
+    def succ
+      @last_prime = @last_prime ? EratosthenesSieve.instance.next_to(@last_prime) : 2
+    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
+    end
+    
+    def succ
+      TrialDivision.instance[@index += 1]
+    end
+    def rewind
+      initialize
+    end
+    alias next succ
+  end
+
+  # Generates all integer which are greater than 2 and
+  # are not divided by 2 nor 3.
+  #
+  # This is a pseudo-prime generator, suitable on 
+  # checking primality of a integer by brute force 
+  # method.
+  class Generator23<PseudoPrimeGenerator
+    def initialize
+      @prime = 1
+      @step = nil
+    end
+    
+    def succ
+      loop do
+	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
+    end
+    alias next succ
+    def rewind
+      initialize
+    end
+  end
+
+
+
+
+  # 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
+
+  # An implementation of eratosthenes's sieve
+  class EratosthenesSieve
+    include Singleton
+
+    def initialize # :nodoc:
+      # bitmap for odd prime numbers less than 256.
+      # For an arbitrary odd number n, @table[i][j] is 1 when n is prime where i,j = n.divmod(32) .
+      @table = [0xcb6e, 0x64b4, 0x129a, 0x816d, 0x4c32, 0x864a, 0x820d, 0x2196]
+    end
+
+    # returns the least odd prime number which is greater than +n+.
+    def next_to(n)
+      n = (n-1).div(2)*2+3 # the next odd number of given n
+      i,j = n.divmod(32)
+      loop do
+	extend_table until @table.length > i
+	if !@table[i].zero?
+	  (j...32).step(2) do |j|
+	    return 32*i+j if !@table[i][j.div(2)].zero?
+	  end
+	end
+	i += 1; j = 1
+      end
+    end
+
+    private
+    def extend_table
+      orig_len = @table.length
+      new_len = [orig_len**2, orig_len+256].min
+      lbound = orig_len*32
+      ubound = new_len*32
+      @table.fill(0xFFFF, orig_len...new_len)
+      (3..Integer(Math.sqrt(ubound))).step(2) do |p|
+	i, j = p.divmod(32)
+	next if @table[i][j.div(2)].zero?
+
+	start = (lbound.div(2*p)*2+1)*p    # odd multiple of p which is greater than or equal to lbound
+	(start...ubound).step(2*p) do |n|
+	  i, j = n.divmod(32)
+	  @table[i] &= 0xFFFF ^ (1<<(j.div(2)))
+	end
+      end
+    end
+  end
+end