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-rw-r--r--lib/prime.rb668
1 files changed, 334 insertions, 334 deletions
diff --git a/lib/prime.rb b/lib/prime.rb
index 8d8598b9e16..d1164dbd052 100644
--- a/lib/prime.rb
+++ b/lib/prime.rb
@@ -99,397 +99,397 @@ class Prime
def method_added(method) # :nodoc:
(class<< self;self;end).def_delegator :instance, method
+ end
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
-# yields all prime numbers p <= +ubound+.
-#
-# == Note
-# +Prime+.+new+ returns a object extended by +Prime+::+OldCompatibility+
-# in order to compatibility to 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
+ # 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
+ # yields all prime numbers p <= +ubound+.
+ #
+ # == Note
+ # +Prime+.+new+ returns a object extended by +Prime+::+OldCompatibility+
+ # in order to compatibility to 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
+ # 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
-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
- if value < 0
- value = -value
- pv = [[-1, 1]]
- else
- pv = []
+ # 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
- for prime in generator
- count = 0
- while (value1, mod = value.divmod(prime)
- mod) == 0
- value = value1
- count += 1
+
+ # 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
+ 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 count != 0
- pv.push [prime, count]
+ if value > 1
+ pv.push [value, 1]
end
- break if value1 <= prime
+ return pv
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
+ # 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 initialize(ubound = nil)
+ @ubound = ubound
+ end
- def upper_bound=(ubound)
- @ubound = ubound
- end
- def upper_bound
- @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
+ # 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
+ # 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
+ # 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.
- 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)
+ # Iterates the given block for each prime numbers.
+ 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
- end
- # see +Enumerator+#with_index.
- alias with_index each_with_index
+ # 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
+ # 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
-end
-# An implementation of +PseudoPrimeGenerator+.
-#
-# Uses +EratosthenesSieve+.
-class EratosthenesGenerator < PseudoPrimeGenerator
- def initialize
- @last_prime = nil
- super
- end
+ # An implementation of +PseudoPrimeGenerator+.
+ #
+ # Uses +EratosthenesSieve+.
+ class EratosthenesGenerator < PseudoPrimeGenerator
+ def initialize
+ @last_prime = nil
+ super
+ end
- def succ
- @last_prime = @last_prime ? EratosthenesSieve.instance.next_to(@last_prime) : 2
- end
- def rewind
- initialize
+ def succ
+ @last_prime = @last_prime ? EratosthenesSieve.instance.next_to(@last_prime) : 2
+ end
+ def rewind
+ initialize
+ end
+ alias next succ
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
+ # 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
+ def succ
+ TrialDivision.instance[@index += 1]
+ end
+ def rewind
+ initialize
+ end
+ alias next succ
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
- super
- 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
+ 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
+ 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
- return @prime
+ end
+ alias next succ
+ def rewind
+ initialize
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.
+ # Internal use. An implementation of prime table by trial division method.
+ class TrialDivision
+ include Singleton
- # 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
+ 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.
- # Returns the cached prime numbers.
- def cache
- return @primes
- end
- alias primes cache
- alias primes_so_far cache
+ # 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
+ # 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
- # 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
+ return @primes[index]
end
- return @primes[index]
end
-end
-# Internal use. An implementation of eratosthenes's sieve
-class EratosthenesSieve
- include Singleton
-
- BITS_PER_ENTRY = 16 # each entry is a set of 16-bits in a Fixnum
- NUMS_PER_ENTRY = BITS_PER_ENTRY * 2 # twiced because even numbers are omitted
- ENTRIES_PER_TABLE = 8
- NUMS_PER_TABLE = NUMS_PER_ENTRY * ENTRIES_PER_TABLE
- FILLED_ENTRY = (1 << NUMS_PER_ENTRY) - 1
-
- def initialize # :nodoc:
- # bitmap for odd prime numbers less than 256.
- # For an arbitrary odd number n, @tables[i][j][k] is
- # * 1 if n is prime,
- # * 0 if n is composite,
- # where i,j,k = indices(n)
- @tables = [[0xcb6e, 0x64b4, 0x129a, 0x816d, 0x4c32, 0x864a, 0x820d, 0x2196].freeze]
- end
+ # Internal use. An implementation of eratosthenes's sieve
+ class EratosthenesSieve
+ include Singleton
- # 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 to given n
- table_index, integer_index, bit_index = indices(n)
- loop do
- extend_table until @tables.length > table_index
- for j in integer_index...ENTRIES_PER_TABLE
- if !@tables[table_index][j].zero?
- for k in bit_index...BITS_PER_ENTRY
- return NUMS_PER_TABLE*table_index + NUMS_PER_ENTRY*j + 2*k+1 if !@tables[table_index][j][k].zero?
+ BITS_PER_ENTRY = 16 # each entry is a set of 16-bits in a Fixnum
+ NUMS_PER_ENTRY = BITS_PER_ENTRY * 2 # twiced because even numbers are omitted
+ ENTRIES_PER_TABLE = 8
+ NUMS_PER_TABLE = NUMS_PER_ENTRY * ENTRIES_PER_TABLE
+ FILLED_ENTRY = (1 << NUMS_PER_ENTRY) - 1
+
+ def initialize # :nodoc:
+ # bitmap for odd prime numbers less than 256.
+ # For an arbitrary odd number n, @tables[i][j][k] is
+ # * 1 if n is prime,
+ # * 0 if n is composite,
+ # where i,j,k = indices(n)
+ @tables = [[0xcb6e, 0x64b4, 0x129a, 0x816d, 0x4c32, 0x864a, 0x820d, 0x2196].freeze]
+ 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 to given n
+ table_index, integer_index, bit_index = indices(n)
+ loop do
+ extend_table until @tables.length > table_index
+ for j in integer_index...ENTRIES_PER_TABLE
+ if !@tables[table_index][j].zero?
+ for k in bit_index...BITS_PER_ENTRY
+ return NUMS_PER_TABLE*table_index + NUMS_PER_ENTRY*j + 2*k+1 if !@tables[table_index][j][k].zero?
+ end
end
+ bit_index = 0
end
- bit_index = 0
+ table_index += 1; integer_index = 0
end
- table_index += 1; integer_index = 0
end
- end
- private
- # for an odd number +n+, returns (i, j, k) such that @tables[i][j][k] represents primarity of the number
- def indices(n)
- # binary digits of n: |0|1|2|3|4|5|6|7|8|9|10|11|....
- # indices: |-| k | j | i
- # because of NUMS_PER_ENTRY, NUMS_PER_TABLE
-
- k = (n & 0b00011111) >> 1
- j = (n & 0b11100000) >> 5
- i = n >> 8
- return i, j, k
- end
+ private
+ # for an odd number +n+, returns (i, j, k) such that @tables[i][j][k] represents primarity of the number
+ def indices(n)
+ # binary digits of n: |0|1|2|3|4|5|6|7|8|9|10|11|....
+ # indices: |-| k | j | i
+ # because of NUMS_PER_ENTRY, NUMS_PER_TABLE
+
+ k = (n & 0b00011111) >> 1
+ j = (n & 0b11100000) >> 5
+ i = n >> 8
+ return i, j, k
+ end
- def extend_table
- lbound = NUMS_PER_TABLE * @tables.length
- ubound = lbound + NUMS_PER_TABLE
- new_table = [FILLED_ENTRY] * ENTRIES_PER_TABLE # which represents primarity in lbound...ubound
- (3..Integer(Math.sqrt(ubound))).step(2) do |p|
- i, j, k = indices(p)
- next if @tables[i][j][k].zero?
-
- start = (lbound.div(p)+1)*p # least multiple of p which is >= lbound
- start += p if start.even?
- (start...ubound).step(2*p) do |n|
- _, j, k = indices(n)
- new_table[j] &= FILLED_ENTRY^(1<<k)
+ def extend_table
+ lbound = NUMS_PER_TABLE * @tables.length
+ ubound = lbound + NUMS_PER_TABLE
+ new_table = [FILLED_ENTRY] * ENTRIES_PER_TABLE # which represents primarity in lbound...ubound
+ (3..Integer(Math.sqrt(ubound))).step(2) do |p|
+ i, j, k = indices(p)
+ next if @tables[i][j][k].zero?
+
+ start = (lbound.div(p)+1)*p # least multiple of p which is >= lbound
+ start += p if start.even?
+ (start...ubound).step(2*p) do |n|
+ _, j, k = indices(n)
+ new_table[j] &= FILLED_ENTRY^(1<<k)
+ end
end
+ @tables << new_table.freeze
end
- @tables << new_table.freeze
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
+ # 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
-end