






'''simple docstring'''







from __future__ import annotations

from math import gcd
def 						A__			(				__lowerCAmelCase		:			Optional[int]   ,    __lowerCAmelCase		:			Union[str, Any] = 2   ,    __lowerCAmelCase		:			Optional[int] = 1   ,    __lowerCAmelCase		:			List[Any] = 3   ,    ):





						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		:			Any   ,    __lowerCAmelCase		:			Dict   ,    __lowerCAmelCase		:			Union[str, Any]						) -> int:
												return (pow(A_   ,    2						) + step) % modulus

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

												while True:
																		# At each iteration, the tortoise moves one step and the hare moves two.
																		lowerCamelCase__         =  rand_fn(A_   ,    A_   ,    A_						)
																		lowerCamelCase__         =  rand_fn(A_   ,    A_   ,    A_						)
																		lowerCamelCase__         =  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``.
																		lowerCamelCase__         =  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.
												lowerCamelCase__         =  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    :      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',
						)
						UpperCamelCase    :      Dict  			=		parser.parse_args()

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









'''simple docstring'''







from manim import *




class 		UpperCamelCase__    (a			):




		'''simple docstring'''





		def 			UpperCamelCase_					(      self       ):
								lowerCamelCase__         =  Rectangle(height=0.5   ,width=0.5       )
								lowerCamelCase__         =  Rectangle(height=0.46   ,width=0.46       ).set_stroke(width=0       )

								lowerCamelCase__         =  [mem.copy() for i in range(6       )]
								lowerCamelCase__         =  [mem.copy() for i in range(6       )]
								lowerCamelCase__         =  VGroup(*_lowerCAmelCase       ).arrange(_lowerCAmelCase   ,buff=0       )
								lowerCamelCase__         =  VGroup(*_lowerCAmelCase       ).arrange(_lowerCAmelCase   ,buff=0       )
								lowerCamelCase__         =  VGroup(_lowerCAmelCase   ,_lowerCAmelCase       ).arrange(_lowerCAmelCase   ,buff=0       )
								lowerCamelCase__         =  Text("""CPU"""   ,font_size=24       )
								lowerCamelCase__         =  Group(_lowerCAmelCase   ,_lowerCAmelCase       ).arrange(_lowerCAmelCase   ,buff=0.5   ,aligned_edge=_lowerCAmelCase       )
								cpu.move_to([-2.5, -0.5, 0]       )
								self.add(_lowerCAmelCase       )

								lowerCamelCase__         =  [mem.copy() for i in range(1       )]
								lowerCamelCase__         =  VGroup(*_lowerCAmelCase       ).arrange(_lowerCAmelCase   ,buff=0       )
								lowerCamelCase__         =  Text("""GPU"""   ,font_size=24       )
								lowerCamelCase__         =  Group(_lowerCAmelCase   ,_lowerCAmelCase       ).arrange(_lowerCAmelCase   ,buff=0.5   ,aligned_edge=_lowerCAmelCase       )
								gpu.align_to(_lowerCAmelCase   ,_lowerCAmelCase       )
								gpu.set_x(gpu.get_x() - 1       )

								self.add(_lowerCAmelCase       )

								lowerCamelCase__         =  [mem.copy() for i in range(6       )]
								lowerCamelCase__         =  VGroup(*_lowerCAmelCase       ).arrange(_lowerCAmelCase   ,buff=0       )

								lowerCamelCase__         =  Text("""Model"""   ,font_size=24       )
								lowerCamelCase__         =  Group(_lowerCAmelCase   ,_lowerCAmelCase       ).arrange(_lowerCAmelCase   ,buff=0.5   ,aligned_edge=_lowerCAmelCase       )
								model.move_to([3, -1.0, 0]       )

								self.play(
								    Create(_lowerCAmelCase   ,run_time=1       )   ,Create(_lowerCAmelCase   ,run_time=1       )   ,Create(_lowerCAmelCase   ,run_time=1       )   ,)

								lowerCamelCase__         =  MarkupText(
								    F'''First, an empty model skeleton is loaded\ninto <span fgcolor=\'{YELLOW}\'>memory</span> without using much RAM.'''   ,font_size=24   ,)

								lowerCamelCase__         =  Square(side_length=2.2       )
								key.move_to([-5, 2, 0]       )

								lowerCamelCase__         =  MarkupText(
								    F'''<b>Key:</b>\n\n<span fgcolor=\'{YELLOW}\'>●</span> Empty Model'''   ,font_size=18   ,)

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

								step_a.move_to([2, 2, 0]       )
								self.play(Write(_lowerCAmelCase   ,run_time=2.5       )   ,Write(_lowerCAmelCase       )   ,Write(_lowerCAmelCase       )       )

								self.add(_lowerCAmelCase       )

								lowerCamelCase__         =  []
								lowerCamelCase__         =  []
								lowerCamelCase__         =  []
								for i, rect in enumerate(_lowerCAmelCase       ):
														lowerCamelCase__         =  Rectangle(height=0.46   ,width=0.46       ).set_stroke(width=0.0       ).set_fill(_lowerCAmelCase   ,opacity=0.7       )
														cpu_target.move_to(_lowerCAmelCase       )
														cpu_target.generate_target()
														lowerCamelCase__         =  0.46 / 4
														lowerCamelCase__         =  0.46 / 3

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

														first_animations.append(rect.animate(run_time=0.5       ).set_stroke(_lowerCAmelCase       )       )
														second_animations.append(MoveToTarget(_lowerCAmelCase   ,run_time=1.5       )       )

								self.play(*_lowerCAmelCase       )
								self.play(*_lowerCAmelCase       )

								self.wait()

