





from __future__ import annotations

from math import gcd



def        __lowercase       (				_SCREAMING_SNAKE_CASE			,						_SCREAMING_SNAKE_CASE = 2			,						_SCREAMING_SNAKE_CASE = 1			,						_SCREAMING_SNAKE_CASE = 3			,						)			->       int | None:


      '''simple docstring'''


      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(_SCREAMING_SNAKE_CASE			,						_SCREAMING_SNAKE_CASE			,						_SCREAMING_SNAKE_CASE     ) -> int:
            return (pow(_SCREAMING_SNAKE_CASE			,						2     ) + step) % modulus

      for _ in range(_SCREAMING_SNAKE_CASE     ):
            # These track the position within the cycle detection logic.
            SCREAMING_SNAKE_CASE		=						seed
            SCREAMING_SNAKE_CASE		=						seed

            while True:
                  # At each iteration, the tortoise moves one step and the hare moves two.
                  SCREAMING_SNAKE_CASE		=						rand_fn(_SCREAMING_SNAKE_CASE			,						_SCREAMING_SNAKE_CASE			,						_SCREAMING_SNAKE_CASE     )
                  SCREAMING_SNAKE_CASE		=						rand_fn(_SCREAMING_SNAKE_CASE			,						_SCREAMING_SNAKE_CASE			,						_SCREAMING_SNAKE_CASE     )
                  SCREAMING_SNAKE_CASE		=						rand_fn(_SCREAMING_SNAKE_CASE			,						_SCREAMING_SNAKE_CASE			,						_SCREAMING_SNAKE_CASE     )

                  # 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``.
                  SCREAMING_SNAKE_CASE		=						gcd(hare - tortoise			,						_SCREAMING_SNAKE_CASE     )

                  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.
            SCREAMING_SNAKE_CASE		=						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

      SCREAMING_SNAKE_CASE_						       =	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""",
      )
      SCREAMING_SNAKE_CASE_						       =	parser.parse_args()

      SCREAMING_SNAKE_CASE_						       =	pollard_rho(args.num, attempts=args.attempts)
      if divisor is None:
            print(F'''{args.num} is probably prime''')
      else:
            SCREAMING_SNAKE_CASE_						       =	args.num // divisor
            print(F'''{args.num} = {divisor} * {quotient}''')









from typing import TYPE_CHECKING

from ...utils import OptionalDependencyNotAvailable, _LazyModule, is_torch_available


SCREAMING_SNAKE_CASE_						       =	{
    """configuration_megatron_bert""": ["""MEGATRON_BERT_PRETRAINED_CONFIG_ARCHIVE_MAP""", """MegatronBertConfig"""],
}

try:
      if not is_torch_available():
            raise OptionalDependencyNotAvailable()
except OptionalDependencyNotAvailable:
      pass
else:
      SCREAMING_SNAKE_CASE_						       =	[
          """MEGATRON_BERT_PRETRAINED_MODEL_ARCHIVE_LIST""",
          """MegatronBertForCausalLM""",
          """MegatronBertForMaskedLM""",
          """MegatronBertForMultipleChoice""",
          """MegatronBertForNextSentencePrediction""",
          """MegatronBertForPreTraining""",
          """MegatronBertForQuestionAnswering""",
          """MegatronBertForSequenceClassification""",
          """MegatronBertForTokenClassification""",
          """MegatronBertModel""",
          """MegatronBertPreTrainedModel""",
      ]

if TYPE_CHECKING:
      from .configuration_megatron_bert import MEGATRON_BERT_PRETRAINED_CONFIG_ARCHIVE_MAP, MegatronBertConfig

      try:
            if not is_torch_available():
                  raise OptionalDependencyNotAvailable()
      except OptionalDependencyNotAvailable:
            pass
      else:
            from .modeling_megatron_bert import (
                MEGATRON_BERT_PRETRAINED_MODEL_ARCHIVE_LIST,
                MegatronBertForCausalLM,
                MegatronBertForMaskedLM,
                MegatronBertForMultipleChoice,
                MegatronBertForNextSentencePrediction,
                MegatronBertForPreTraining,
                MegatronBertForQuestionAnswering,
                MegatronBertForSequenceClassification,
                MegatronBertForTokenClassification,
                MegatronBertModel,
                MegatronBertPreTrainedModel,
            )

else:
      import sys

      SCREAMING_SNAKE_CASE_						       =	_LazyModule(__name__, globals()["""__file__"""], _import_structure, module_spec=__spec__)


