
"""simple docstring"""



import math


def  _UpperCamelCase	(						_A      )    ->  bool:





    """simple docstring"""

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

    return exponent == int(_A      )


def  _UpperCamelCase	(						_A = 1 / 1_2_3_4_5      )    ->  int:





    """simple docstring"""

    _UpperCAmelCase          =   0
    _UpperCAmelCase          =   0

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


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

"""simple docstring"""



import colorsys

from PIL import Image  # type: ignore


def  _UpperCamelCase	(						_A					,      _A					,      _A      )    ->  float:





    """simple docstring"""

    _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  _UpperCamelCase	(						_A      )    ->  tuple:





    """simple docstring"""

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


def  _UpperCamelCase	(						_A      )    ->  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(_A					,      1					,      1      )      )


def  _UpperCamelCase	(						_A = 8_0_0					,      _A = 6_0_0					,      _A = -0.6					,      _A = 0					,      _A = 3.2					,      _A = 5_0					,      _A = True					,      )    ->  Image.Image:





    """simple docstring"""

    _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
 a					:	List[str]							=		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()