


'''simple docstring'''






from __future__ import annotations

from math import gcd






def 		_A			(lowerCAmelCase__					:int			,  lowerCAmelCase__					:int = 2			,  lowerCAmelCase__					:int = 1			,  lowerCAmelCase__					:int = 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(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.
          _a      =							seed
          _a      =							seed

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

 a_   :      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",
 )
 a_   :      Tuple				    =      parser.parse_args()

 a_   :      Optional[Any]				    =      pollard_rho(args.num, attempts=args.attempts)
 if divisor is None:
  print(f'''{args.num} is probably prime''')
 else:
  a_   :      Optional[Any]				    =      args.num // divisor
  print(f'''{args.num} = {divisor} * {quotient}''')





'''simple docstring'''






def 		_A			(lowerCAmelCase__					:int						)				->					int:




     '''simple docstring'''



     assert (
         isinstance(lowerCAmelCase__			,  lowerCAmelCase__						) and number_of_steps > 0
     ), f'number_of_steps needs to be positive integer, your input {number_of_steps}'
     if number_of_steps == 1:
          return 1
     _a ,				_a      =							1, 1
     for _ in range(number_of_steps - 1						):
          _a ,				_a      =							current + previous, current
     return current


if __name__ == "__main__":
 import doctest

 doctest.testmod()

