



'''simple docstring'''




from __future__ import annotations

from math import gcd



def      a      (			lowerCamelCase__						,				lowerCamelCase__ = 2						,				lowerCamelCase__ = 1						,				lowerCamelCase__ = 3						,				):
			'''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__						,				lowerCamelCase__						,				lowerCamelCase__   ) -> int:
						return (pow(lowerCamelCase__						,				2   ) + step) % modulus

			for _ in range(lowerCamelCase__   ):
						# These track the position within the cycle detection logic.
						A_     :						Tuple     =					seed
						A_     :						Tuple     =					seed

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

			lowerCamelCase							:Optional[int]       = pollard_rho(args.num, attempts=args.attempts)
			if divisor is None:
						print(F"{args.num} is probably prime")
			else:
						lowerCamelCase							:Tuple       = args.num // divisor
						print(F"{args.num} = {divisor} * {quotient}")




'''simple docstring'''




from __future__ import annotations

from functools import lru_cache
from math import ceil

lowerCamelCase							:List[str]       = 1_0_0

lowerCamelCase							:Dict       = set(range(3, NUM_PRIMES, 2))
primes.add(2)
lowerCamelCase							:int

for prime in range(3, ceil(NUM_PRIMES**0.5), 2):
						if prime not in primes:
												continue
						primes.difference_update(set(range(prime * prime, NUM_PRIMES, prime)))



@lru_cache(maxsize=1_00   )
def      a      (			lowerCamelCase__   ):
			'''simple docstring'''





			if number_to_partition < 0:
						return set()
			elif number_to_partition == 0:
						return {1}

			A_     :						set[int]     =					set()
			A_     :						int
			A_     :						int

			for prime in primes:
						if prime > number_to_partition:
									continue
						for sub in partition(number_to_partition - prime   ):
									ret.add(sub * prime   )

			return ret



def      a      (			lowerCamelCase__ = 50_00   ):
			'''simple docstring'''





			for number_to_partition in range(1						,				lowerCamelCase__   ):
						if len(partition(lowerCamelCase__   )   ) > number_unique_partitions:
									return number_to_partition
			return None


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