





from __future__ import annotations

from math import gcd

def    __snake_case  (    _lowerCAmelCase       :       List[Any] , _lowerCAmelCase       :       Tuple = 2 , _lowerCAmelCase       :       Any = 1 , _lowerCAmelCase       :       Tuple = 3 , )       ->  List[str]:
   # 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       :       Optional[int] , _lowerCAmelCase       :       Optional[Any] , _lowerCAmelCase       :       Optional[Any]		) -> int:
      return (pow(lowerCAmelCase__ , 2		) + step) % modulus

   for _ in range(lowerCAmelCase__		):
      # These track the position within the cycle detection logic.
      A_    : str								=	seed
      A_    : Optional[Any]								=	seed

      while True:
         # At each iteration, the tortoise moves one step and the hare moves two.
         A_    : List[Any]								=	rand_fn(lowerCAmelCase__ , lowerCAmelCase__ , lowerCAmelCase__		)
         A_    : Optional[Any]								=	rand_fn(lowerCAmelCase__ , lowerCAmelCase__ , lowerCAmelCase__		)
         A_    : int								=	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_    : Any								=	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_    : str								=	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							:					Union[str, 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							:					Any		    =		parser.parse_args()

 _lowerCAmelCase							:					str		    =		pollard_rho(args.num, attempts=args.attempts)
 if divisor is None:
  print(F'''{args.num} is probably prime''')
 else:
  _lowerCAmelCase							:					Union[str, Any]		    =		args.num // divisor
  print(F'''{args.num} = {divisor} * {quotient}''')


import random

from .binary_exp_mod import bin_exp_mod



def        UpperCamelCase    (							lowerCAmelCase__ ,			lowerCAmelCase__=1000		):
					'''simple docstring'''
					if n < 2:
										return False

					if n % 2 == 0:
										return n == 2

					# this means n is odd
					lowercase							=						n - 1
					lowercase							=						0
					while d % 2 == 0:
										d /= 2
										exp += 1

					# n - 1=d*(2**exp)
					lowercase							=						0
					while count < prec:
										lowercase							=						random.randint(2 ,			n - 1		)
										lowercase							=						bin_exp_mod(lowerCAmelCase__ ,			lowerCAmelCase__ ,			lowerCAmelCase__		)
										if b != 1:
															lowercase							=						True
															for _ in range(lowerCAmelCase__		):
																				if b == n - 1:
																									lowercase							=						False
																									break
																				lowercase							=						b * b
																				b %= n
															if flag:
																				return False
															count += 1
					return True


if __name__ == "__main__":
							lowercase__		:Tuple   =    abs(int(input("Enter bound : ").strip()))
							print("Here's the list of primes:")
							print(", ".join(str(i) for i in range(n + 1) if is_prime_big(i)))




