

'''simple docstring'''





from __future__ import annotations

from math import gcd







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

					for _ in range(__UpperCamelCase	):
										# 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(__UpperCamelCase    ,__UpperCamelCase    ,__UpperCamelCase	)
															lowerCamelCase_  				=      rand_fn(__UpperCamelCase    ,__UpperCamelCase    ,__UpperCamelCase	)
															lowerCamelCase_  				=      rand_fn(__UpperCamelCase    ,__UpperCamelCase    ,__UpperCamelCase	)

															# 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    ,__UpperCamelCase	)

															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

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

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





'''simple docstring'''
import colorsys

from PIL import Image  # type: ignore







def 		A_(     A						:		float		,     A						:		float		,     A						:		int):
							UpperCamelCase   			=     x
							UpperCamelCase   			=     y
							for step in range(A):  # noqa: B007
														UpperCamelCase   			=     a * a - b * b + x
														UpperCamelCase   			=     2 * a * b + y
														UpperCamelCase   			=     a_new

														# divergence happens for all complex number with an absolute value
														# greater than 4
														if a * a + b * b > 4:
																					break
							return step / (max_step - 1)







def 		A_(     A						:		float):
							if distance == 1:
														return (0, 0, 0)
							else:
														return (255, 255, 255)







def 		A_(     A						:		float):
							if distance == 1:
														return (0, 0, 0)
							else:
														return tuple(round(i * 255) for i in colorsys.hsv_to_rgb(A		,     1		,     1))







def 		A_(     A						:		int = 800		,     A						:		int = 600		,     A						:		float = -0.6		,     A						:		float = 0		,     A						:		float = 3.2		,     A						:		int = 50		,     A						:		bool = True		,     ):
							UpperCamelCase   			=     Image.new('RGB'		,     (image_width, image_height))
							UpperCamelCase   			=     img.load()

							# loop through the image-coordinates
							for image_x in range(A):
														for image_y in range(A):
																					# determine the figure-coordinates based on the image-coordinates
																					UpperCamelCase   			=     figure_width / image_width * image_height
																					UpperCamelCase   			=     figure_center_x + (image_x / image_width - 0.5) * figure_width
																					UpperCamelCase   			=     figure_center_y + (image_y / image_height - 0.5) * figure_height

																					UpperCamelCase   			=     get_distance(A		,     A		,     A)

																					# color the corresponding pixel based on the selected coloring-function
																					if use_distance_color_coding:
																												UpperCamelCase   			=     get_color_coded_rgb(A)
																					else:
																												UpperCamelCase   			=     get_black_and_white_rgb(A)

							return img


if __name__ == "__main__":
						import doctest

						doctest.testmod()

						# colored version, full figure
						lowerCAmelCase					:    Any           = get_image()

						# uncomment for colored version, different section, zoomed in
						# img = get_image(figure_center_x = -0.6, figure_center_y = -0.4,
						# figure_width = 0.8)

						# uncomment for black and white version, full figure
						# img = get_image(use_distance_color_coding = False)

						# uncomment to save the image
						# img.save("mandelbrot.png")

						img.show()




