


"""simple docstring"""


from __future__ import annotations

from math import gcd




def   snake_case		(     UpperCamelCase__       :						int			,				UpperCamelCase__       :						int = 2			,				UpperCamelCase__       :						int = 1			,				UpperCamelCase__       :						int = 3			,				)   ->      int | None:
      # A value less than 2 can cause an infinite loop in the algorithm.
      if num < 2:
            raise ValueError("""The input value cannot be less than 2"""    )

      # Because of the relationship between ``f(f(x))`` and ``f(x)``, this
      # algorithm struggles to find factors that are divisible by two.
      # As a workaround, we specifically check for two and even inputs.
      #   See: https://math.stackexchange.com/a/2856214/165820
      if num > 2 and num % 2 == 0:
            return 2

      # Pollard's Rho algorithm requires a function that returns pseudorandom
      # values between 0 <= X < ``num``.  It doesn't need to be random in the
      # sense that the output value is cryptographically secure or difficult
      # to calculate, it only needs to be random in the sense that all output
      # values should be equally likely to appear.
      # For this reason, Pollard suggested using ``f(x) = (x**2 - 1) % num``
      # However, the success of Pollard's algorithm isn't guaranteed and is
      # determined in part by the initial seed and the chosen random function.
      # To make retries easier, we will instead use ``f(x) = (x**2 + C) % num``
      # where ``C`` is a value that we can modify between each attempt.
      def rand_fn(UpperCamelCase__       :						int			,				UpperCamelCase__       :						int			,				UpperCamelCase__       :						int    ) -> int:
            return (pow(UpperCamelCase__			,				2    ) + step) % modulus

      for _ in range(UpperCamelCase__    ):
            # These track the position within the cycle detection logic.
            lowerCamelCase					:					List[Any]        =	seed
            lowerCamelCase					:					Dict        =	seed

            while True:
                  # At each iteration, the tortoise moves one step and the hare moves two.
                  lowerCamelCase					:					Tuple        =	rand_fn(UpperCamelCase__			,				UpperCamelCase__			,				UpperCamelCase__    )
                  lowerCamelCase					:					Any        =	rand_fn(UpperCamelCase__			,				UpperCamelCase__			,				UpperCamelCase__    )
                  lowerCamelCase					:					int        =	rand_fn(UpperCamelCase__			,				UpperCamelCase__			,				UpperCamelCase__    )

                  # At some point both the tortoise and the hare will enter a cycle whose
                  # length ``p`` is a divisor of ``num``.  Once in that cycle, at some point
                  # the tortoise and hare will end up on the same value modulo ``p``.
                  # We can detect when this happens because the position difference between
                  # the tortoise and the hare will share a common divisor with ``num``.
                  lowerCamelCase					:					List[Any]        =	gcd(hare - tortoise			,				UpperCamelCase__    )

                  if divisor == 1:
                        # No common divisor yet, just keep searching.
                        continue
                  else:
                        # We found a common divisor!
                        if divisor == num:
                              # Unfortunately, the divisor is ``num`` itself and is useless.
                              break
                        else:
                              # The divisor is a nontrivial factor of ``num``!
                              return divisor

        # If we made it here, then this attempt failed.
        # We need to pick a new starting seed for the tortoise and hare
        # in addition to a new step value for the random function.
        # To keep this example implementation deterministic, the
        # new values will be generated based on currently available
        # values instead of using something like ``random.randint``.

        # We can use the hare's position as the new seed.
        # This is actually what Richard Brent's the "optimized" variant does.
            lowerCamelCase					:					Optional[int]        =	hare

            # The new step value for the random function can just be incremented.
            # At first the results will be similar to what the old function would
            # have produced, but the value will quickly diverge after a bit.
            step += 1

      # We haven't found a divisor within the requested number of attempts.
      # We were unlucky or ``num`` itself is actually prime.
      return None


if __name__ == "__main__":
    import argparse

    __lowerCamelCase      :Optional[Any]	    =		argparse.ArgumentParser()
    parser.add_argument(
        'num',
        type=int,
        help='The value to find a divisor of',
    )
    parser.add_argument(
        '--attempts',
        type=int,
        default=3,
        help='The number of attempts before giving up',
    )
    __lowerCamelCase      :Tuple	    =		parser.parse_args()

    __lowerCamelCase      :Dict	    =		pollard_rho(args.num, attempts=args.attempts)
    if divisor is None:
        print(f"""{args.num} is probably prime""")
    else:
        __lowerCamelCase      :Any	    =		args.num // divisor
        print(f"""{args.num} = {divisor} * {quotient}""")








"""simple docstring"""


from __future__ import annotations

__lowerCamelCase      :int	    =		10




def   snake_case		(     UpperCamelCase__       :						list[int]    )   ->      list[int]:
      lowerCamelCase					:					int        =	1
      lowerCamelCase					:					Union[str, Any]        =	max(UpperCamelCase__    )
      while placement <= max_digit:
            # declare and initialize empty buckets
            lowerCamelCase					:					list[list]        =	[[] for _ in range(UpperCamelCase__    )]
            # split list_of_ints between the buckets
            for i in list_of_ints:
                  lowerCamelCase					:					Any        =	int((i / placement) % RADIX    )
                  buckets[tmp].append(UpperCamelCase__    )
            # put each buckets' contents into list_of_ints
            lowerCamelCase					:					Dict        =	0
            for b in range(UpperCamelCase__    ):
                  for i in buckets[b]:
                        lowerCamelCase					:					List[str]        =	i
                        a += 1
        # move to next
            placement *= RADIX
      return list_of_ints


if __name__ == "__main__":
    import doctest

    doctest.testmod()




