





'''simple docstring'''




from __future__ import annotations

from math import gcd





def 	__lowerCamelCase			(      __lowerCAmelCase       :							int      ,    __lowerCAmelCase       :							int = 2      ,    __lowerCAmelCase       :							int = 1      ,    __lowerCAmelCase       :							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(__lowerCAmelCase       :							int      ,    __lowerCAmelCase       :							int      ,    __lowerCAmelCase       :							int		) -> int:
      return (pow(__lowerCAmelCase      ,    2		) + step) % modulus

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

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

         # 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``.
         snake_case 						=   gcd(hare - tortoise      ,    __lowerCAmelCase		)

         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.
      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}""")








'''simple docstring'''




import os
from collections.abc import Iterator





def 	__lowerCamelCase			(      __lowerCAmelCase       :							str = "."		)		->  Iterator[str]:
   for dir_path, dir_names, filenames in os.walk(__lowerCAmelCase		):
      snake_case 						=   [d for d in dir_names if d != """scripts""" and d[0] not in """._"""]
      for filename in filenames:
         if filename == "__init__.py":
            continue
         if os.path.splitext(__lowerCAmelCase		)[1] in (".py", ".ipynb"):
            yield os.path.join(__lowerCAmelCase      ,    __lowerCAmelCase		).lstrip("""./"""		)





def 	__lowerCamelCase			(      __lowerCAmelCase       :							List[str]		)		->  Any:
   return F'''{i * "  "}*''' if i else "\n##"





def 	__lowerCamelCase			(      __lowerCAmelCase       :							str      ,    __lowerCAmelCase       :							str		)		->  str:
   snake_case 						=   old_path.split(os.sep		)
   for i, new_part in enumerate(new_path.split(os.sep		)		):
      if (i + 1 > len(__lowerCAmelCase		) or old_parts[i] != new_part) and new_part:
         print(F'''{md_prefix(__lowerCAmelCase		)} {new_part.replace("_"      ,    " "		).title()}'''		)
   return new_path





def 	__lowerCamelCase			(      __lowerCAmelCase       :							str = "."		)		->  None:
   snake_case 						=   """"""
   for filepath in sorted(good_file_paths(__lowerCAmelCase		)		):
      snake_case						,    snake_case 						=   os.path.split(__lowerCAmelCase		)
      if filepath != old_path:
         snake_case 						=   print_path(__lowerCAmelCase      ,    __lowerCAmelCase		)
      snake_case 						=   (filepath.count(os.sep		) + 1) if filepath else 0
      snake_case 						=   F'''{filepath}/{filename}'''.replace(""" """      ,    """%20"""		)
      snake_case 						=   os.path.splitext(filename.replace("""_"""      ,    """ """		).title()		)[0]
      print(F'''{md_prefix(__lowerCAmelCase		)} [{filename}]({url})'''		)


if __name__ == "__main__":
 print_directory_md(".")

