"""simple docstring"""






import colorsys

from PIL import Image  # type: ignore


def      UpperCAmelCase__			(							lowerCAmelCase__	:float   ,						lowerCAmelCase__	:float   ,						lowerCAmelCase__	:int						)					->       float:
      '''simple docstring'''


      lowercase								=      x
      lowercase								=      y
      for step in range(lowerCAmelCase__						):  # noqa: B007
            lowercase								=      a * a - b * b + x
            lowercase								=      2 * a * b + y
            lowercase								=      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      UpperCAmelCase__			(							lowerCAmelCase__	:float						)					->       tuple:
      '''simple docstring'''


      if distance == 1:
            return (0, 0, 0)
      else:
            return (2_5_5, 2_5_5, 2_5_5)


def      UpperCAmelCase__			(							lowerCAmelCase__	:float						)					->       tuple:
      '''simple docstring'''


      if distance == 1:
            return (0, 0, 0)
      else:
            return tuple(round(i * 2_5_5						) for i in colorsys.hsv_to_rgb(lowerCAmelCase__   ,						1   ,						1						)						)


def      UpperCAmelCase__			(							lowerCAmelCase__	:int = 8_0_0   ,						lowerCAmelCase__	:int = 6_0_0   ,						lowerCAmelCase__	:float = -0.6   ,						lowerCAmelCase__	:float = 0   ,						lowerCAmelCase__	:float = 3.2   ,						lowerCAmelCase__	:int = 5_0   ,						lowerCAmelCase__	:bool = True   ,						)					->       Image.Image:
      '''simple docstring'''


      lowercase								=      Image.new("""RGB"""   ,						(image_width, image_height)						)
      lowercase								=      img.load()

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

                  lowercase								=      get_distance(lowerCAmelCase__   ,						lowerCAmelCase__   ,						lowerCAmelCase__						)

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

      return img


if __name__ == "__main__":
       import doctest

       doctest.testmod()

       # colored version, full figure
       __lowerCAmelCase       :			Tuple   	=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()




"""simple docstring"""






import math


def      UpperCAmelCase__			(							lowerCAmelCase__	:int						)					->       bool:
      '''simple docstring'''


      lowercase								=      math.loga(math.sqrt(4 * positive_integer + 1						) / 2 + 1 / 2						)

      return exponent == int(lowerCAmelCase__						)


def      UpperCAmelCase__			(							lowerCAmelCase__	:float = 1 / 1_2_3_4_5						)					->       int:
      '''simple docstring'''


      lowercase								=      0
      lowercase								=      0

      lowercase								=      3
      while True:
            lowercase								=      (integer**2 - 1) / 4
            # if candidate is an integer, then there is a partition for k
            if partition_candidate == int(lowerCAmelCase__						):
                  lowercase								=      int(lowerCAmelCase__						)
                  total_partitions += 1
                  if check_partition_perfect(lowerCAmelCase__						):
                        perfect_partitions += 1
            if perfect_partitions > 0:
                  if perfect_partitions / total_partitions < max_proportion:
                        return int(lowerCAmelCase__						)
            integer += 1


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



