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Edit File : prime.rb \| Size : 13.20 KB
# # = 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 # # 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 [2, 3, 5, 7, 11] # # == Retrieving the instance # # +Prime+.new is obsolete.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 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) value = -value if value < 0 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 = primeindex } 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_1e_1 * p_2*e_2 .... * p_ne_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(&block) return self.dup unless block if @ubound last_value = nil loop do prime = succ break last_value if prime > @ubound last_value = block.call(prime) 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_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 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 # 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(&block) return @generator.dup unless block_given? loop do yield succ end end end end Back

Edit File : prime.rb | Size : 13.20 KB

#
# = 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
#
# 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 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)
    value = -value if value < 0
    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(&block)
      return self.dup unless block
      if @ubound
        last_value = nil
        loop do
          prime = succ
          break last_value if prime > @ubound
          last_value = block.call(prime)
        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_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
      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

# 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(&block)
      return @generator.dup unless block_given?
      loop do
        yield succ
      end
    end
  end
end

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