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561 lines
15 KiB
Ruby
561 lines
15 KiB
Ruby
# frozen_string_literal: false
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#
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# = prime.rb
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#
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# Prime numbers and factorization library.
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#
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# Copyright::
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# Copyright (c) 1998-2008 Keiju ISHITSUKA(SHL Japan Inc.)
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# Copyright (c) 2008 Yuki Sonoda (Yugui) <yugui@yugui.jp>
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#
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# Documentation::
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# Yuki Sonoda
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#
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require "singleton"
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require "forwardable"
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class Integer
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# Re-composes a prime factorization and returns the product.
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#
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# See Prime#int_from_prime_division for more details.
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def Integer.from_prime_division(pd)
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Prime.int_from_prime_division(pd)
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end
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# Returns the factorization of +self+.
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#
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# See Prime#prime_division for more details.
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def prime_division(generator = Prime::Generator23.new)
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Prime.prime_division(self, generator)
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end
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# Returns true if +self+ is a prime number, else returns false.
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# Not recommended for very big integers (> 10**23).
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def prime?
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return self >= 2 if self <= 3
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if (bases = miller_rabin_bases)
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return miller_rabin_test(bases)
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end
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return true if self == 5
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return false unless 30.gcd(self) == 1
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(7..Integer.sqrt(self)).step(30) do |p|
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return false if
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self%(p) == 0 || self%(p+4) == 0 || self%(p+6) == 0 || self%(p+10) == 0 ||
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self%(p+12) == 0 || self%(p+16) == 0 || self%(p+22) == 0 || self%(p+24) == 0
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end
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true
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end
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MILLER_RABIN_BASES = [
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[2],
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[2,3],
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[31,73],
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[2,3,5],
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[2,3,5,7],
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[2,7,61],
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[2,13,23,1662803],
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[2,3,5,7,11],
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[2,3,5,7,11,13],
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[2,3,5,7,11,13,17],
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[2,3,5,7,11,13,17,19,23],
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[2,3,5,7,11,13,17,19,23,29,31,37],
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[2,3,5,7,11,13,17,19,23,29,31,37,41],
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].map!(&:freeze).freeze
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private_constant :MILLER_RABIN_BASES
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private def miller_rabin_bases
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# Miller-Rabin's complexity is O(k log^3n).
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# So we can reduce the complexity by reducing the number of bases tested.
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# Using values from https://en.wikipedia.org/wiki/Miller%E2%80%93Rabin_primality_test
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i = case
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when self < 0xffff then
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# For small integers, Miller Rabin can be slower
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# There is no mathematical significance to 0xffff
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return nil
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# when self < 2_047 then 0
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when self < 1_373_653 then 1
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when self < 9_080_191 then 2
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when self < 25_326_001 then 3
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when self < 3_215_031_751 then 4
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when self < 4_759_123_141 then 5
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when self < 1_122_004_669_633 then 6
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when self < 2_152_302_898_747 then 7
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when self < 3_474_749_660_383 then 8
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when self < 341_550_071_728_321 then 9
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when self < 3_825_123_056_546_413_051 then 10
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when self < 318_665_857_834_031_151_167_461 then 11
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when self < 3_317_044_064_679_887_385_961_981 then 12
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else return nil
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end
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MILLER_RABIN_BASES[i]
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end
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private def miller_rabin_test(bases)
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return false if even?
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r = 0
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d = self >> 1
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while d.even?
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d >>= 1
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r += 1
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end
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self_minus_1 = self-1
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bases.each do |a|
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x = a.pow(d, self)
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next if x == 1 || x == self_minus_1 || a == self
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return false if r.times do
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x = x.pow(2, self)
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break if x == self_minus_1
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end
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end
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true
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end
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# Iterates the given block over all prime numbers.
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#
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# See +Prime+#each for more details.
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def Integer.each_prime(ubound, &block) # :yields: prime
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Prime.each(ubound, &block)
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end
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end
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#
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# The set of all prime numbers.
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#
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# == Example
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#
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# Prime.each(100) do |prime|
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# p prime #=> 2, 3, 5, 7, 11, ...., 97
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# end
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#
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# Prime is Enumerable:
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#
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# Prime.first 5 # => [2, 3, 5, 7, 11]
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#
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# == Retrieving the instance
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#
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# For convenience, each instance method of +Prime+.instance can be accessed
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# as a class method of +Prime+.
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#
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# e.g.
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# Prime.instance.prime?(2) #=> true
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# Prime.prime?(2) #=> true
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#
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# == Generators
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#
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# A "generator" provides an implementation of enumerating pseudo-prime
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# numbers and it remembers the position of enumeration and upper bound.
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# Furthermore, it is an external iterator of prime enumeration which is
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# compatible with an Enumerator.
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#
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# +Prime+::+PseudoPrimeGenerator+ is the base class for generators.
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# There are few implementations of generator.
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#
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# [+Prime+::+EratosthenesGenerator+]
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# Uses Eratosthenes' sieve.
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# [+Prime+::+TrialDivisionGenerator+]
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# Uses the trial division method.
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# [+Prime+::+Generator23+]
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# Generates all positive integers which are not divisible by either 2 or 3.
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# This sequence is very bad as a pseudo-prime sequence. But this
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# is faster and uses much less memory than the other generators. So,
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# it is suitable for factorizing an integer which is not large but
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# has many prime factors. e.g. for Prime#prime? .
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class Prime
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VERSION = "0.1.2"
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include Enumerable
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include Singleton
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class << self
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extend Forwardable
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include Enumerable
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def method_added(method) # :nodoc:
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(class<< self;self;end).def_delegator :instance, method
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end
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end
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# Iterates the given block over all prime numbers.
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#
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# == Parameters
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#
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# +ubound+::
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# Optional. An arbitrary positive number.
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# The upper bound of enumeration. The method enumerates
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# prime numbers infinitely if +ubound+ is nil.
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# +generator+::
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# Optional. An implementation of pseudo-prime generator.
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#
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# == Return value
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#
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# An evaluated value of the given block at the last time.
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# Or an enumerator which is compatible to an +Enumerator+
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# if no block given.
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#
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# == Description
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#
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# Calls +block+ once for each prime number, passing the prime as
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# a parameter.
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#
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# +ubound+::
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# Upper bound of prime numbers. The iterator stops after it
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# yields all prime numbers p <= +ubound+.
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#
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def each(ubound = nil, generator = EratosthenesGenerator.new, &block)
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generator.upper_bound = ubound
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generator.each(&block)
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end
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# Returns true if +obj+ is an Integer and is prime. Also returns
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# true if +obj+ is a Module that is an ancestor of +Prime+.
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# Otherwise returns false.
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def include?(obj)
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case obj
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when Integer
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prime?(obj)
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when Module
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Module.instance_method(:include?).bind(Prime).call(obj)
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else
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false
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end
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end
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# Returns true if +value+ is a prime number, else returns false.
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# Integer#prime? is much more performant.
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#
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# == Parameters
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#
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# +value+:: an arbitrary integer to be checked.
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# +generator+:: optional. A pseudo-prime generator.
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def prime?(value, generator = Prime::Generator23.new)
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raise ArgumentError, "Expected a prime generator, got #{generator}" unless generator.respond_to? :each
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raise ArgumentError, "Expected an integer, got #{value}" unless value.respond_to?(:integer?) && value.integer?
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return false if value < 2
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generator.each do |num|
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q,r = value.divmod num
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return true if q < num
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return false if r == 0
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end
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end
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# Re-composes a prime factorization and returns the product.
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#
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# For the decomposition:
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#
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# [[p_1, e_1], [p_2, e_2], ..., [p_n, e_n]],
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#
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# it returns:
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#
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# p_1**e_1 * p_2**e_2 * ... * p_n**e_n.
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#
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# == Parameters
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# +pd+:: Array of pairs of integers.
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# Each pair consists of a prime number -- a prime factor --
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# and a natural number -- its exponent (multiplicity).
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#
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# == Example
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# Prime.int_from_prime_division([[3, 2], [5, 1]]) #=> 45
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# 3**2 * 5 #=> 45
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#
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def int_from_prime_division(pd)
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pd.inject(1){|value, (prime, index)|
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value * prime**index
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}
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end
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# Returns the factorization of +value+.
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#
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# For an arbitrary integer:
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#
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# p_1**e_1 * p_2**e_2 * ... * p_n**e_n,
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#
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# prime_division returns an array of pairs of integers:
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#
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# [[p_1, e_1], [p_2, e_2], ..., [p_n, e_n]].
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#
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# Each pair consists of a prime number -- a prime factor --
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# and a natural number -- its exponent (multiplicity).
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#
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# == Parameters
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# +value+:: An arbitrary integer.
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# +generator+:: Optional. A pseudo-prime generator.
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# +generator+.succ must return the next
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# pseudo-prime number in ascending order.
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# It must generate all prime numbers,
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# but may also generate non-prime numbers, too.
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#
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# === Exceptions
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# +ZeroDivisionError+:: when +value+ is zero.
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#
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# == Example
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#
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# Prime.prime_division(45) #=> [[3, 2], [5, 1]]
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# 3**2 * 5 #=> 45
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#
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def prime_division(value, generator = Prime::Generator23.new)
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raise ZeroDivisionError if value == 0
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if value < 0
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value = -value
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pv = [[-1, 1]]
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else
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pv = []
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end
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generator.each do |prime|
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count = 0
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while (value1, mod = value.divmod(prime)
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mod) == 0
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value = value1
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count += 1
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end
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if count != 0
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pv.push [prime, count]
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end
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break if value1 <= prime
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end
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if value > 1
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pv.push [value, 1]
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end
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pv
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end
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# An abstract class for enumerating pseudo-prime numbers.
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#
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# Concrete subclasses should override succ, next, rewind.
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class PseudoPrimeGenerator
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include Enumerable
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def initialize(ubound = nil)
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@ubound = ubound
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end
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def upper_bound=(ubound)
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@ubound = ubound
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end
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def upper_bound
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@ubound
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end
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# returns the next pseudo-prime number, and move the internal
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# position forward.
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#
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# +PseudoPrimeGenerator+#succ raises +NotImplementedError+.
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def succ
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raise NotImplementedError, "need to define `succ'"
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end
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# alias of +succ+.
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def next
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raise NotImplementedError, "need to define `next'"
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end
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# Rewinds the internal position for enumeration.
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#
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# See +Enumerator+#rewind.
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def rewind
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raise NotImplementedError, "need to define `rewind'"
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end
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# Iterates the given block for each prime number.
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def each
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return self.dup unless block_given?
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if @ubound
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last_value = nil
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loop do
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prime = succ
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break last_value if prime > @ubound
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last_value = yield prime
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end
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else
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loop do
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yield succ
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end
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end
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end
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# see +Enumerator+#with_index.
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def with_index(offset = 0, &block)
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return enum_for(:with_index, offset) { Float::INFINITY } unless block
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return each_with_index(&block) if offset == 0
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each do |prime|
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yield prime, offset
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offset += 1
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end
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end
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# see +Enumerator+#with_object.
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def with_object(obj)
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return enum_for(:with_object, obj) { Float::INFINITY } unless block_given?
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each do |prime|
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yield prime, obj
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end
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end
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def size
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Float::INFINITY
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end
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end
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# An implementation of +PseudoPrimeGenerator+.
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#
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# Uses +EratosthenesSieve+.
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class EratosthenesGenerator < PseudoPrimeGenerator
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def initialize
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@last_prime_index = -1
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super
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end
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def succ
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@last_prime_index += 1
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EratosthenesSieve.instance.get_nth_prime(@last_prime_index)
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end
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def rewind
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initialize
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end
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alias next succ
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end
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# An implementation of +PseudoPrimeGenerator+ which uses
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# a prime table generated by trial division.
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class TrialDivisionGenerator < PseudoPrimeGenerator
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def initialize
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@index = -1
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super
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end
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def succ
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TrialDivision.instance[@index += 1]
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end
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def rewind
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initialize
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end
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alias next succ
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end
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# Generates all integers which are greater than 2 and
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# are not divisible by either 2 or 3.
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#
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# This is a pseudo-prime generator, suitable on
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# checking primality of an integer by brute force
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# method.
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class Generator23 < PseudoPrimeGenerator
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def initialize
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@prime = 1
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@step = nil
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super
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end
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def succ
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if (@step)
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@prime += @step
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@step = 6 - @step
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else
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case @prime
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when 1; @prime = 2
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when 2; @prime = 3
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when 3; @prime = 5; @step = 2
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end
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end
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@prime
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end
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alias next succ
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def rewind
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initialize
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end
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end
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# Internal use. An implementation of prime table by trial division method.
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class TrialDivision
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include Singleton
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def initialize # :nodoc:
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# These are included as class variables to cache them for later uses. If memory
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# usage is a problem, they can be put in Prime#initialize as instance variables.
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# There must be no primes between @primes[-1] and @next_to_check.
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@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]
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# @next_to_check % 6 must be 1.
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@next_to_check = 103 # @primes[-1] - @primes[-1] % 6 + 7
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@ulticheck_index = 3 # @primes.index(@primes.reverse.find {|n|
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# n < Math.sqrt(@@next_to_check) })
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@ulticheck_next_squared = 121 # @primes[@ulticheck_index + 1] ** 2
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end
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# Returns the +index+th prime number.
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#
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# +index+ is a 0-based index.
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def [](index)
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while index >= @primes.length
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# Only check for prime factors up to the square root of the potential primes,
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# but without the performance hit of an actual square root calculation.
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if @next_to_check + 4 > @ulticheck_next_squared
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@ulticheck_index += 1
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@ulticheck_next_squared = @primes.at(@ulticheck_index + 1) ** 2
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end
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# Only check numbers congruent to one and five, modulo six. All others
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# are divisible by two or three. This also allows us to skip checking against
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# two and three.
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@primes.push @next_to_check if @primes[2..@ulticheck_index].find {|prime| @next_to_check % prime == 0 }.nil?
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@next_to_check += 4
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@primes.push @next_to_check if @primes[2..@ulticheck_index].find {|prime| @next_to_check % prime == 0 }.nil?
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@next_to_check += 2
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end
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@primes[index]
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end
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end
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# Internal use. An implementation of Eratosthenes' sieve
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class EratosthenesSieve
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include Singleton
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def initialize
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@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]
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# @max_checked must be an even number
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@max_checked = @primes.last + 1
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end
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def get_nth_prime(n)
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compute_primes while @primes.size <= n
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@primes[n]
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end
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private
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def compute_primes
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# max_segment_size must be an even number
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max_segment_size = 1e6.to_i
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max_cached_prime = @primes.last
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# do not double count primes if #compute_primes is interrupted
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# by Timeout.timeout
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@max_checked = max_cached_prime + 1 if max_cached_prime > @max_checked
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segment_min = @max_checked
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segment_max = [segment_min + max_segment_size, max_cached_prime * 2].min
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root = Integer.sqrt(segment_max)
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segment = ((segment_min + 1) .. segment_max).step(2).to_a
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|
|
(1..Float::INFINITY).each do |sieving|
|
|
prime = @primes[sieving]
|
|
break if prime > root
|
|
composite_index = (-(segment_min + 1 + prime) / 2) % prime
|
|
while composite_index < segment.size do
|
|
segment[composite_index] = nil
|
|
composite_index += prime
|
|
end
|
|
end
|
|
|
|
@primes.concat(segment.compact!)
|
|
|
|
@max_checked = segment_max
|
|
end
|
|
end
|
|
end
|