



from __future__ import annotations

from math import gcd






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

							for _ in range(a__       ):
														# 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(a__							,       a__							,       a__       )
																					__SCREAMING_SNAKE_CASE									= rand_fn(a__							,       a__							,       a__       )
																					__SCREAMING_SNAKE_CASE									= rand_fn(a__							,       a__							,       a__       )

																					# 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							,       a__       )

																					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__		:					Tuple							=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__		:					List[Any]							=parser.parse_args()

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











from __future__ import annotations

from collections.abc import Generator






def      __lowercase       (			)    ->			Generator[int, None, None]:
							__SCREAMING_SNAKE_CASE									= {}
							__SCREAMING_SNAKE_CASE									= 2
							while True:
														__SCREAMING_SNAKE_CASE									= factor_map.pop(a__							,       a__       )
														if factor:
																					__SCREAMING_SNAKE_CASE									= factor + prime
																					while x in factor_map:
																												x += factor
																					__SCREAMING_SNAKE_CASE									= factor
														else:
																					__SCREAMING_SNAKE_CASE									= prime
																					yield prime
														prime += 1






def      __lowercase       (			a__ = 1E10       )    ->			int:
							__SCREAMING_SNAKE_CASE									= sieve()

							__SCREAMING_SNAKE_CASE									= 1
							while True:
														__SCREAMING_SNAKE_CASE									= next(a__       )
														if (2 * prime * n) > limit:
																					return n
														# Ignore the next prime as the reminder will be 2.
														next(a__       )
														n += 2


if __name__ == "__main__":
			print(solution())






