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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, 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# +Prime+.new is obsolete.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