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# frozen_string_literal: false
#
# = 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.
# Not recommended for very big integers (> 10**23).
def prime?
return self >= 2 if self <= 3
if (bases = miller_rabin_bases)
return miller_rabin_test(bases)
end
return true if self == 5
return false unless 30.gcd(self) == 1
(7..Integer.sqrt(self)).step(30) do |p|
return false if
self%(p) == 0 || self%(p+4) == 0 || self%(p+6) == 0 || self%(p+10) == 0 ||
self%(p+12) == 0 || self%(p+16) == 0 || self%(p+22) == 0 || self%(p+24) == 0
end
true
end
MILLER_RABIN_BASES = [
[2],
[2,3],
[31,73],
[2,3,5],
[2,3,5,7],
[2,7,61],
[2,13,23,1662803],
[2,3,5,7,11],
[2,3,5,7,11,13],
[2,3,5,7,11,13,17],
[2,3,5,7,11,13,17,19,23],
[2,3,5,7,11,13,17,19,23,29,31,37],
[2,3,5,7,11,13,17,19,23,29,31,37,41],
].map!(&:freeze).freeze
private_constant :MILLER_RABIN_BASES
private def miller_rabin_bases
# Miller-Rabin's complexity is O(k log^3n).
# So we can reduce the complexity by reducing the number of bases tested.
# Using values from https://en.wikipedia.org/wiki/Miller%E2%80%93Rabin_primality_test
i = case
when self < 0xffff then
# For small integers, Miller Rabin can be slower
# There is no mathematical significance to 0xffff
return nil
# when self < 2_047 then 0
when self < 1_373_653 then 1
when self < 9_080_191 then 2
when self < 25_326_001 then 3
when self < 3_215_031_751 then 4
when self < 4_759_123_141 then 5
when self < 1_122_004_669_633 then 6
when self < 2_152_302_898_747 then 7
when self < 3_474_749_660_383 then 8
when self < 341_550_071_728_321 then 9
when self < 3_825_123_056_546_413_051 then 10
when self < 318_665_857_834_031_151_167_461 then 11
when self < 3_317_044_064_679_887_385_961_981 then 12
else return nil
end
MILLER_RABIN_BASES[i]
end
private def miller_rabin_test(bases)
return false if even?
r = 0
d = self >> 1
while d.even?
d >>= 1
r += 1
end
self_minus_1 = self-1
bases.each do |a|
x = a.pow(d, self)
next if x == 1 || x == self_minus_1 || a == self
return false if r.times do
x = x.pow(2, self)
break if x == self_minus_1
end
end
true
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
#
# 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
VERSION = "0.1.2"
include Enumerable
include Singleton
class << self
extend Forwardable
include Enumerable
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+.
#
def each(ubound = nil, generator = EratosthenesGenerator.new, &block)
generator.upper_bound = ubound
generator.each(&block)
end
# Returns true if +obj+ is an Integer and is prime. Also returns
# true if +obj+ is a Module that is an ancestor of +Prime+.
# Otherwise returns false.
def include?(obj)
case obj
when Integer
prime?(obj)
when Module
Module.instance_method(:include?).bind(Prime).call(obj)
else
false
end
end
# Returns true if +value+ is a prime number, else returns false.
# Integer#prime? is much more performant.
#
# == Parameters
#
# +value+:: an arbitrary integer to be checked.
# +generator+:: optional. A pseudo-prime generator.
def prime?(value, generator = Prime::Generator23.new)
raise ArgumentError, "Expected a prime generator, got #{generator}" unless generator.respond_to? :each
raise ArgumentError, "Expected an integer, got #{value}" unless value.respond_to?(:integer?) && value.integer?
return false if value < 2
generator.each do |num|
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.
#
# For the decomposition:
#
# [[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.
#
# == Parameters
# +pd+:: Array of pairs of integers.
# Each pair consists of a prime number -- a prime factor --
# and a natural number -- its exponent (multiplicity).
#
# == Example
# Prime.int_from_prime_division([[3, 2], [5, 1]]) #=> 45
# 3**2 * 5 #=> 45
#
def int_from_prime_division(pd)
pd.inject(1){|value, (prime, index)|
value * prime**index
}
end
# Returns the factorization of +value+.
#
# For an arbitrary integer:
#
# p_1**e_1 * p_2**e_2 * ... * p_n**e_n,
#
# prime_division returns an array of pairs of integers:
#
# [[p_1, e_1], [p_2, e_2], ..., [p_n, e_n]].
#
# Each pair consists of a prime number -- a prime factor --
# and a natural number -- its exponent (multiplicity).
#
# == Parameters
# +value+:: An arbitrary integer.
# +generator+:: Optional. A pseudo-prime generator.
# +generator+.succ must return the next
# pseudo-prime number in ascending order.
# It must generate all prime numbers,
# but may also generate non-prime numbers, too.
#
# === Exceptions
# +ZeroDivisionError+:: when +value+ is zero.
#
# == Example
#
# Prime.prime_division(45) #=> [[3, 2], [5, 1]]
# 3**2 * 5 #=> 45
#
def prime_division(value, generator = Prime::Generator23.new)
raise ZeroDivisionError if value == 0
if value < 0
value = -value
pv = [[-1, 1]]
else
pv = []
end
generator.each do |prime|
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
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.
def with_index(offset = 0, &block)
return enum_for(:with_index, offset) { Float::INFINITY } unless block
return each_with_index(&block) if offset == 0
each do |prime|
yield prime, offset
offset += 1
end
end
# see +Enumerator+#with_object.
def with_object(obj)
return enum_for(:with_object, obj) { Float::INFINITY } unless block_given?
each do |prime|
yield prime, obj
end
end
def size
Float::INFINITY
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
@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 +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
@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.sqrt(segment_max)
segment = ((segment_min + 1) .. segment_max).step(2).to_a
(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