
"""simple docstring"""






import random





def __A						(   a_ :							int     )->							bool:




		'''simple docstring'''






		SCREAMING_SNAKE_CASE       :     Optional[Any]				= num - 1
		SCREAMING_SNAKE_CASE       :     str				= 0

		while s % 2 == 0:
				SCREAMING_SNAKE_CASE       :     Union[str, Any]				= s // 2
				t += 1

		for _ in range(5     ):
				SCREAMING_SNAKE_CASE       :     Any				= random.randrange(2		,		num - 1     )
				SCREAMING_SNAKE_CASE       :     Optional[int]				= pow(a_		,		a_		,		a_     )
				if v != 1:
						SCREAMING_SNAKE_CASE       :     Union[str, Any]				= 0
						while v != (num - 1):
								if i == t - 1:
										return False
								else:
										SCREAMING_SNAKE_CASE       :     int				= i + 1
										SCREAMING_SNAKE_CASE       :     int				= (v**2) % num
		return True





def __A						(   a_ :							int     )->							bool:




		'''simple docstring'''






		if num < 2:
				return False

		SCREAMING_SNAKE_CASE       :     List[str]				= [
		    2,
		    3,
		    5,
		    7,
		    11,
		    13,
		    17,
		    19,
		    23,
		    29,
		    31,
		    37,
		    41,
		    43,
		    47,
		    53,
		    59,
		    61,
		    67,
		    71,
		    73,
		    79,
		    83,
		    89,
		    97,
		    1_01,
		    1_03,
		    1_07,
		    1_09,
		    1_13,
		    1_27,
		    1_31,
		    1_37,
		    1_39,
		    1_49,
		    1_51,
		    1_57,
		    1_63,
		    1_67,
		    1_73,
		    1_79,
		    1_81,
		    1_91,
		    1_93,
		    1_97,
		    1_99,
		    2_11,
		    2_23,
		    2_27,
		    2_29,
		    2_33,
		    2_39,
		    2_41,
		    2_51,
		    2_57,
		    2_63,
		    2_69,
		    2_71,
		    2_77,
		    2_81,
		    2_83,
		    2_93,
		    3_07,
		    3_11,
		    3_13,
		    3_17,
		    3_31,
		    3_37,
		    3_47,
		    3_49,
		    3_53,
		    3_59,
		    3_67,
		    3_73,
		    3_79,
		    3_83,
		    3_89,
		    3_97,
		    4_01,
		    4_09,
		    4_19,
		    4_21,
		    4_31,
		    4_33,
		    4_39,
		    4_43,
		    4_49,
		    4_57,
		    4_61,
		    4_63,
		    4_67,
		    4_79,
		    4_87,
		    4_91,
		    4_99,
		    5_03,
		    5_09,
		    5_21,
		    5_23,
		    5_41,
		    5_47,
		    5_57,
		    5_63,
		    5_69,
		    5_71,
		    5_77,
		    5_87,
		    5_93,
		    5_99,
		    6_01,
		    6_07,
		    6_13,
		    6_17,
		    6_19,
		    6_31,
		    6_41,
		    6_43,
		    6_47,
		    6_53,
		    6_59,
		    6_61,
		    6_73,
		    6_77,
		    6_83,
		    6_91,
		    7_01,
		    7_09,
		    7_19,
		    7_27,
		    7_33,
		    7_39,
		    7_43,
		    7_51,
		    7_57,
		    7_61,
		    7_69,
		    7_73,
		    7_87,
		    7_97,
		    8_09,
		    8_11,
		    8_21,
		    8_23,
		    8_27,
		    8_29,
		    8_39,
		    8_53,
		    8_57,
		    8_59,
		    8_63,
		    8_77,
		    8_81,
		    8_83,
		    8_87,
		    9_07,
		    9_11,
		    9_19,
		    9_29,
		    9_37,
		    9_41,
		    9_47,
		    9_53,
		    9_67,
		    9_71,
		    9_77,
		    9_83,
		    9_91,
		    9_97,
		]

		if num in low_primes:
				return True

		for prime in low_primes:
				if (num % prime) == 0:
						return False

		return rabin_miller(a_     )





def __A						(   a_ :							int = 10_24     )->							int:




		'''simple docstring'''






		while True:
				SCREAMING_SNAKE_CASE       :     Tuple				= random.randrange(2 ** (keysize - 1)		,		2 ** (keysize)     )
				if is_prime_low_num(a_     ):
						return num


if __name__ == "__main__":
					lowerCamelCase__ :       Union[str, Any]						     =			generate_large_prime()
					print(("Prime number:", num))
					print(("is_prime_low_num:", is_prime_low_num(num)))







"""simple docstring"""






from __future__ import annotations

from numpy import array, cos, cross, floataa, radians, sin
from numpy.typing import NDArray





def __A						(   a_ :							float		,		a_ :							float		,		a_ :							bool = False     )->							list[float]:




		'''simple docstring'''






		if radian_mode:
				return [magnitude * cos(a_     ), magnitude * sin(a_     )]
		return [magnitude * cos(radians(a_     )     ), magnitude * sin(radians(a_     )     )]





def __A						(   a_ :							NDArray[floataa]		,		a_ :							NDArray[floataa]		,		a_ :							float = 10**-1     )->							bool:




		'''simple docstring'''






		SCREAMING_SNAKE_CASE       :     NDArray[floataa]				= cross(a_		,		a_     )
		SCREAMING_SNAKE_CASE       :     float				= sum(a_     )
		return abs(a_     ) < eps


if __name__ == "__main__":
					# Test to check if it works
					lowerCamelCase__ :       Optional[Any]						     =			array(
					    [
					        polar_force(7_1_8.4, 180 - 30),
					        polar_force(8_7_9.5_4, 45),
					        polar_force(100, -90),
					    ]
					)

					lowerCamelCase__ :       NDArray[floataa]						     =			array([[0, 0], [0, 0], [0, 0]])

					assert in_static_equilibrium(forces, location)

					# Problem 1 in image_data/2D_problems.jpg
					lowerCamelCase__ :       Union[str, Any]						     =			array(
					    [
					        polar_force(30 * 9.8_1, 15),
					        polar_force(215, 180 - 45),
					        polar_force(264, 90 - 30),
					    ]
					)

					lowerCamelCase__ :       Any						     =			array([[0, 0], [0, 0], [0, 0]])

					assert in_static_equilibrium(forces, location)

					# Problem in image_data/2D_problems_1.jpg
					lowerCamelCase__ :       Union[str, Any]						     =			array([[0, -2000], [0, -1200], [0, 15600], [0, -12400]])

					lowerCamelCase__ :       Optional[int]						     =			array([[0, 0], [6, 0], [10, 0], [12, 0]])

					assert in_static_equilibrium(forces, location)

					import doctest

					doctest.testmod()





