




import random







def      _a				(						SCREAMING_SNAKE_CASE_							:    str					):
					__lowerCAmelCase           =	num - 1
					__lowerCAmelCase           =	0

					while s % 2 == 0:
										__lowerCAmelCase           =	s // 2
										t += 1

					for _ in range(5					):
										__lowerCAmelCase           =	random.randrange(2				,				num - 1					)
										__lowerCAmelCase           =	pow(_lowercase				,				_lowercase				,				_lowercase					)
										if v != 1:
															__lowerCAmelCase           =	0
															while v != (num - 1):
																				if i == t - 1:
																									return False
																				else:
																									__lowerCAmelCase           =	i + 1
																									__lowerCAmelCase           =	(v**2) % num
					return True







def      _a				(						SCREAMING_SNAKE_CASE_							:    Optional[Any]					):







					if num < 2:
										return False

					__lowerCAmelCase           =	[
					    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(_lowercase					)







def      _a				(						SCREAMING_SNAKE_CASE_							:    Optional[int] = 10_24					):







					while True:
										__lowerCAmelCase           =	random.randrange(2 ** (keysize - 1)				,				2 ** (keysize)					)
										if is_prime_low_num(_lowercase					):
															return num


if __name__ == "__main__":
					UpperCamelCase__         =      generate_large_prime()
					print(("""Prime number:""", num))
					print(("""is_prime_low_num:""", is_prime_low_num(num)))












import math







def      _a				(						SCREAMING_SNAKE_CASE_							:    int					):

					if 1 < number < 4:
										# 2 and 3 are primes
										return True
					elif number < 2 or number % 2 == 0 or number % 3 == 0:
										# Negatives, 0, 1, all even numbers, all multiples of 3 are not primes
										return False

					# All primes number are in format of 6k +/- 1
					for i in range(5				,				int(math.sqrt(SCREAMING_SNAKE_CASE_					) + 1					)				,				6					):
										if number % i == 0 or number % (i + 2) == 0:
															return False
					return True







def      _a				(						SCREAMING_SNAKE_CASE_							:    int = 1_00_01					):

					try:
										__lowerCAmelCase           =	int(SCREAMING_SNAKE_CASE_					)
					except (TypeError, ValueError):
										raise TypeError("Parameter nth must be int or castable to int."					) from None
					if nth <= 0:
										raise ValueError("Parameter nth must be greater than or equal to one."					)
					__lowerCAmelCase           =	[]
					__lowerCAmelCase           =	2
					while len(SCREAMING_SNAKE_CASE_					) < nth:
										if is_prime(SCREAMING_SNAKE_CASE_					):
															primes.append(SCREAMING_SNAKE_CASE_					)
															num += 1
										else:
															num += 1
					return primes[len(SCREAMING_SNAKE_CASE_					) - 1]


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




