

'''simple docstring'''
from __future__ import annotations

from math import gcd


def 	_lowerCAmelCase (						_UpperCamelCase						:					int							,			_UpperCamelCase						:					int = 2							,			_UpperCamelCase						:					int = 1							,			_UpperCamelCase						:					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(_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.
										_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(_UpperCamelCase							,			_UpperCamelCase							,			_UpperCamelCase    )
															_SCREAMING_SNAKE_CASE							   =rand_fn(_UpperCamelCase							,			_UpperCamelCase							,			_UpperCamelCase    )
															_SCREAMING_SNAKE_CASE							   =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``.
															_SCREAMING_SNAKE_CASE							   =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.
										_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

					lowerCamelCase			:				str 			=					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			:				Union[str, Any] 			=					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			:				int 			=					args.num // divisor
										print(f'''{args.num} = {divisor} * {quotient}''')



'''simple docstring'''
from __future__ import annotations

from fractions import Fraction
from math import gcd, sqrt


def 	_lowerCAmelCase (						_UpperCamelCase						:					int    )				->      bool:
     """simple docstring"""




     _SCREAMING_SNAKE_CASE							   =int(number**0.5    )
     return number == sq * sq


def 	_lowerCAmelCase (						_UpperCamelCase						:					int							,			_UpperCamelCase						:					int							,			_UpperCamelCase						:					int							,			_UpperCamelCase						:					int							,			_UpperCamelCase						:					int							,			_UpperCamelCase						:					int    )				->      tuple[int, int]:
     """simple docstring"""




     _SCREAMING_SNAKE_CASE							   =x_num * y_den * z_den + y_num * x_den * z_den + z_num * x_den * y_den
     _SCREAMING_SNAKE_CASE							   =x_den * y_den * z_den
     _SCREAMING_SNAKE_CASE							   =gcd(_UpperCamelCase							,			_UpperCamelCase    )
     top //= hcf
     bottom //= hcf
     return top, bottom


def 	_lowerCAmelCase (						_UpperCamelCase						:					int = 35    )				->      int:
     """simple docstring"""




     _SCREAMING_SNAKE_CASE							   =set()
     _SCREAMING_SNAKE_CASE							   =42
     _SCREAMING_SNAKE_CASE							   =Fraction(0    )
     _SCREAMING_SNAKE_CASE							   =42

     for x_num in range(1							,			order + 1    ):
          for x_den in range(x_num + 1							,			order + 1    ):
               for y_num in range(1							,			order + 1    ):
                    for y_den in range(y_num + 1							,			order + 1    ):
                         # n=1
                         _SCREAMING_SNAKE_CASE							   =x_num * y_den + x_den * y_num
                         _SCREAMING_SNAKE_CASE							   =x_den * y_den
                         _SCREAMING_SNAKE_CASE							   =gcd(_UpperCamelCase							,			_UpperCamelCase    )
                         z_num //= hcf
                         z_den //= hcf
                         if 0 < z_num < z_den <= order:
                              _SCREAMING_SNAKE_CASE							   =add_three(
                                  _UpperCamelCase							,			_UpperCamelCase							,			_UpperCamelCase							,			_UpperCamelCase							,			_UpperCamelCase							,			_UpperCamelCase    )
                              unique_s.add(_UpperCamelCase    )

                         # n=2
                         _SCREAMING_SNAKE_CASE							   =(
                             x_num * x_num * y_den * y_den + x_den * x_den * y_num * y_num
                         )
                         _SCREAMING_SNAKE_CASE							   =x_den * x_den * y_den * y_den
                         if is_sq(_UpperCamelCase    ) and is_sq(_UpperCamelCase    ):
                              _SCREAMING_SNAKE_CASE							   =int(sqrt(_UpperCamelCase    )    )
                              _SCREAMING_SNAKE_CASE							   =int(sqrt(_UpperCamelCase    )    )
                              _SCREAMING_SNAKE_CASE							   =gcd(_UpperCamelCase							,			_UpperCamelCase    )
                              z_num //= hcf
                              z_den //= hcf
                              if 0 < z_num < z_den <= order:
                                   _SCREAMING_SNAKE_CASE							   =add_three(
                                       _UpperCamelCase							,			_UpperCamelCase							,			_UpperCamelCase							,			_UpperCamelCase							,			_UpperCamelCase							,			_UpperCamelCase    )
                                   unique_s.add(_UpperCamelCase    )

                    # n=-1
                         _SCREAMING_SNAKE_CASE							   =x_num * y_num
                         _SCREAMING_SNAKE_CASE							   =x_den * y_num + x_num * y_den
                         _SCREAMING_SNAKE_CASE							   =gcd(_UpperCamelCase							,			_UpperCamelCase    )
                         z_num //= hcf
                         z_den //= hcf
                         if 0 < z_num < z_den <= order:
                              _SCREAMING_SNAKE_CASE							   =add_three(
                                  _UpperCamelCase							,			_UpperCamelCase							,			_UpperCamelCase							,			_UpperCamelCase							,			_UpperCamelCase							,			_UpperCamelCase    )
                              unique_s.add(_UpperCamelCase    )

                         # n=2
                         _SCREAMING_SNAKE_CASE							   =x_num * x_num * y_num * y_num
                         _SCREAMING_SNAKE_CASE							   =(
                             x_den * x_den * y_num * y_num + x_num * x_num * y_den * y_den
                         )
                         if is_sq(_UpperCamelCase    ) and is_sq(_UpperCamelCase    ):
                              _SCREAMING_SNAKE_CASE							   =int(sqrt(_UpperCamelCase    )    )
                              _SCREAMING_SNAKE_CASE							   =int(sqrt(_UpperCamelCase    )    )
                              _SCREAMING_SNAKE_CASE							   =gcd(_UpperCamelCase							,			_UpperCamelCase    )
                              z_num //= hcf
                              z_den //= hcf
                              if 0 < z_num < z_den <= order:
                                   _SCREAMING_SNAKE_CASE							   =add_three(
                                       _UpperCamelCase							,			_UpperCamelCase							,			_UpperCamelCase							,			_UpperCamelCase							,			_UpperCamelCase							,			_UpperCamelCase    )
                                   unique_s.add(_UpperCamelCase    )

     for num, den in unique_s:
          total += Fraction(_UpperCamelCase							,			_UpperCamelCase    )

     return total.denominator + total.numerator


if __name__ == "__main__":
     print(f'''{solution() = }''')
