






from __future__ import annotations

from math import gcd




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

	for _ in range(lowercase__  ):
		# These track the position within the cycle detection logic.
		UpperCamelCase__ :Dict						=      seed
		UpperCamelCase__ :List[Any]						=      seed

		while True:
			# At each iteration, the tortoise moves one step and the hare moves two.
			UpperCamelCase__ :Optional[int]						=      rand_fn(lowercase__ ,		lowercase__ ,		lowercase__  )
			UpperCamelCase__ :List[str]						=      rand_fn(lowercase__ ,		lowercase__ ,		lowercase__  )
			UpperCamelCase__ :str						=      rand_fn(lowercase__ ,		lowercase__ ,		lowercase__  )

			# 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``.
			UpperCamelCase__ :str						=      gcd(hare - tortoise ,		lowercase__  )

			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.
		UpperCamelCase__ :Union[str, Any]						=      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

							UpperCamelCase        =				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",
							)
							UpperCamelCase        =				parser.parse_args()

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



'''simple docstring'''
from manim import *




class        _UpperCamelCase	(    SCREAMING_SNAKE_CASE):


				'''simple docstring'''


				def 	a__		(     self							)    ->  List[str]:
					lowercase						:		List[Any]   					=		Rectangle(height=0.5							,      width=0.5							)
					lowercase						:		str   					=		Rectangle(height=0.25							,      width=0.25							)
					lowercase						:		Union[str, Any]   					=		Rectangle(height=0.46							,      width=0.46							).set_stroke(width=0							)

					lowercase						:		List[str]   					=		[mem.copy() for i in range(6							)]
					lowercase						:		Any   					=		[mem.copy() for i in range(6							)]
					lowercase						:		List[str]   					=		VGroup(*a_							).arrange(a_							,      buff=0							)
					lowercase						:		List[Any]   					=		VGroup(*a_							).arrange(a_							,      buff=0							)
					lowercase						:		List[Any]   					=		VGroup(a_							,      a_							).arrange(a_							,      buff=0							)
					lowercase						:		Union[str, Any]   					=		Text("CPU"							,      font_size=2_4							)
					lowercase						:		List[Any]   					=		Group(a_							,      a_							).arrange(a_							,      buff=0.5							,      aligned_edge=a_							)
					cpu.move_to([-2.5, -0.5, 0]							)
					self.add(a_							)

					lowercase						:		List[Any]   					=		[mem.copy() for i in range(4							)]
					lowercase						:		Union[str, Any]   					=		VGroup(*a_							).arrange(a_							,      buff=0							)
					lowercase						:		Dict   					=		Text("GPU"							,      font_size=2_4							)
					lowercase						:		Tuple   					=		Group(a_							,      a_							).arrange(a_							,      buff=0.5							,      aligned_edge=a_							)
					gpu.move_to([-1, -1, 0]							)
					self.add(a_							)

					lowercase						:		Tuple   					=		[mem.copy() for i in range(6							)]
					lowercase						:		Optional[int]   					=		VGroup(*a_							).arrange(a_							,      buff=0							)

					lowercase						:		Any   					=		Text("Model"							,      font_size=2_4							)
					lowercase						:		str   					=		Group(a_							,      a_							).arrange(a_							,      buff=0.5							,      aligned_edge=a_							)
					model.move_to([3, -1.0, 0]							)
					self.add(a_							)

					lowercase						:		Dict   					=		[]
					lowercase						:		Tuple   					=		[]
					lowercase						:		List[Any]   					=		[]

					for i, rect in enumerate(a_							):
						rect.set_stroke(a_							)

						lowercase						:		Tuple   					=		Rectangle(height=0.46 / 4							,      width=0.46 / 3							).set_stroke(width=0.0							).set_fill(a_							,      opacity=0.7							)

						if i == 0:
							cpu_target.next_to(cpu_left_col_base[0].get_corner(DOWN + LEFT							)							,      buff=0.02							,      direction=a_							)
							cpu_target.set_x(cpu_target.get_x() + 0.1							)
						elif i == 3:
							cpu_target.next_to(model_cpu_arr[0]							,      direction=a_							,      buff=0.0							)
						else:
							cpu_target.next_to(model_cpu_arr[i - 1]							,      direction=a_							,      buff=0.0							)
						self.add(a_							)
						model_cpu_arr.append(a_							)

					self.add(*a_							,      *a_							,      *a_							)

					lowercase						:		Any   					=		[mem.copy() for i in range(6							)]
					lowercase						:		Dict   					=		VGroup(*a_							).arrange(a_							,      buff=0							)

					lowercase						:		List[str]   					=		Text("Loaded Checkpoint"							,      font_size=2_4							)
					lowercase						:		Optional[Any]   					=		Group(a_							,      a_							).arrange(a_							,      buff=0.5							,      aligned_edge=a_							)
					checkpoint.move_to([3, 0.5, 0]							)

					self.add(a_							)

					lowercase						:		Any   					=		[]
					lowercase						:		int   					=		[]

					for i, rect in enumerate(a_							):
						lowercase						:		str   					=		fill.copy().set_fill(a_							,      opacity=0.7							)
						target.move_to(a_							)
						ckpt_arr.append(a_							)

						lowercase						:		Optional[Any]   					=		target.copy()
						if i < 5:
							cpu_target.move_to(cpu_left_col_base[i + 1]							)
						else:
							cpu_target.move_to(cpu_right_col_base[i - 5]							)
						ckpt_cpu_arr.append(a_							)
					self.add(*a_							,      *a_							)

					lowercase						:		Optional[Any]   					=		Square(side_length=2.2							)
					key.move_to([-5, 2, 0]							)

					lowercase						:		str   					=		MarkupText(
					    F'''<b>Key:</b>\n\n<span fgcolor=\'{YELLOW}\'>●</span> Empty Model'''							,      font_size=1_8							,      )

					key_text.move_to([-5, 2.4, 0]							)

					self.add(a_							,      a_							)

					lowercase						:		Any   					=		MarkupText(
					    F'''<span fgcolor=\'{BLUE}\'>●</span> Checkpoint'''							,      font_size=1_8							,      )

					blue_text.next_to(a_							,      DOWN * 2.4							,      aligned_edge=key_text.get_left()							)
					self.add(a_							)

					lowercase						:		List[Any]   					=		MarkupText(
					    F'''Based on the passed in configuration, weights are stored in\na variety of np.memmaps on disk or to a particular device.'''							,      font_size=2_4							,      )
					step_a.move_to([2, 2, 0]							)

					lowercase						:		Any   					=		[meta_mem.copy() for i in range(6							)]
					lowercase						:		Dict   					=		[meta_mem.copy() for i in range(6							)]
					lowercase						:		Union[str, Any]   					=		VGroup(*a_							).arrange(a_							,      buff=0							)
					lowercase						:		Dict   					=		VGroup(*a_							).arrange(a_							,      buff=0							)
					lowercase						:		Any   					=		VGroup(a_							,      a_							).arrange(a_							,      buff=0							)
					lowercase						:		Optional[Any]   					=		Text("Disk"							,      font_size=2_4							)
					lowercase						:		List[str]   					=		Group(a_							,      a_							).arrange(a_							,      buff=0.5							,      aligned_edge=a_							)
					disk.move_to([-4.0, -1.25, 0]							)
					self.play(Write(a_							,      run_time=3							)							,      Write(a_							,      run_time=1							)							,      Create(a_							,      run_time=1							)							)

					lowercase						:		Optional[Any]   					=		[]
					for i, rect in enumerate(a_							):
						lowercase						:		int   					=		rect.copy()
						target.generate_target()
						target.target.move_to(disk_left_col_base[i]							).scale(0.5							)
						animations.append(MoveToTarget(a_							,      run_time=1.5							)							)
					self.play(*a_							)

					self.play(FadeOut(a_							)							)

					lowercase						:		List[Any]   					=		MarkupText(F'''Then, the checkpoint is removed from memory\nthrough garbage collection.'''							,      font_size=2_4							)
					step_a.move_to([2, 2, 0]							)

					self.play(Write(a_							,      run_time=3							)							)

					self.play(
					    FadeOut(a_							,      a_							,      *a_							,      *a_							)							,      )

					self.wait()


