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# 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
  include Singleton

  include Enumerable[Integer]

  extend Enumerable[Integer]

  # 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 self?.each: (?Integer? ubound, ?PseudoPrimeGenerator generator) { (Integer) -> void } -> void
                | (?Integer? ubound, ?PseudoPrimeGenerator generator) -> PseudoPrimeGenerator

  # 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 self?.int_from_prime_division: (Array[[ Integer, Integer ]]) -> Integer

  # Returns true if `value` is a prime number, else returns false.
  #
  # ## Parameters
  #
  # `value`
  # :   an arbitrary integer to be checked.
  # `generator`
  # :   optional. A pseudo-prime generator.
  #
  def self?.prime?: (Integer value, ?PseudoPrimeGenerator generator) -> bool

  # 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 self?.prime_division: (Integer, ?PseudoPrimeGenerator generator) -> Array[[ Integer, Integer ]]

  # Returns the singleton instance.
  #
  def self.instance: () -> Prime

  # An abstract class for enumerating pseudo-prime numbers.
  #
  # Concrete subclasses should override succ, next, rewind.
  #
  class PseudoPrimeGenerator
    def initialize: (?Integer?) -> void

    include Enumerable[Integer]

    attr_accessor upper_bound (): Integer?

    # Iterates the given block for each prime number.
    #
    def each: () { (Integer) -> void } -> void

    # alias of `succ`.
    #
    def next: () -> Integer

    # Rewinds the internal position for enumeration.
    #
    # See `Enumerator`#rewind.
    #
    def rewind: () -> void

    def size: () -> Float

    # returns the next pseudo-prime number, and move the internal position forward.
    #
    # `PseudoPrimeGenerator`#succ raises `NotImplementedError`.
    #
    def succ: () -> Integer
  end

  # An implementation of `PseudoPrimeGenerator`.
  #
  # Uses `EratosthenesSieve`.
  #
  class EratosthenesGenerator < PseudoPrimeGenerator
  end

  # An implementation of `PseudoPrimeGenerator` which uses a prime table generated
  # by trial division.
  #
  class TrialDivisionGenerator < PseudoPrimeGenerator
  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
  end
end