






def        _a  (     UpperCamelCase_					:				int    ,   UpperCamelCase_					:				int   ) ->   str:
						"""simple docstring"""







						if not isinstance(UpperCamelCase_    ,   UpperCamelCase_   ):
												raise ValueError("iterations must be defined as integers"   )
						if not isinstance(UpperCamelCase_    ,   UpperCamelCase_   ) or not number >= 1:
												raise ValueError(
												    "starting number must be\n                         and integer and be more than 0"   )
						if not iterations >= 1:
												raise ValueError("Iterations must be done more than 0 times to play FizzBuzz"   )

						lowerCAmelCase__       =						""
						while number <= iterations:
												if number % 3 == 0:
																		out += "Fizz"
												if number % 5 == 0:
																		out += "Buzz"
												if 0 not in (number % 3, number % 5):
																		out += str(UpperCamelCase_   )

												# print(out)
												number += 1
												out += " "
						return out


if __name__ == "__main__":
	import doctest

	doctest.testmod()














import random


def        _a  (     UpperCamelCase_					:				int   ) ->   bool:
						"""simple docstring"""







						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(UpperCamelCase_    ,   UpperCamelCase_    ,   UpperCamelCase_   )
												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  (     UpperCamelCase_					:				int   ) ->   bool:
						"""simple docstring"""







						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,
						    101,
						    103,
						    107,
						    109,
						    113,
						    127,
						    131,
						    137,
						    139,
						    149,
						    151,
						    157,
						    163,
						    167,
						    173,
						    179,
						    181,
						    191,
						    193,
						    197,
						    199,
						    211,
						    223,
						    227,
						    229,
						    233,
						    239,
						    241,
						    251,
						    257,
						    263,
						    269,
						    271,
						    277,
						    281,
						    283,
						    293,
						    307,
						    311,
						    313,
						    317,
						    331,
						    337,
						    347,
						    349,
						    353,
						    359,
						    367,
						    373,
						    379,
						    383,
						    389,
						    397,
						    401,
						    409,
						    419,
						    421,
						    431,
						    433,
						    439,
						    443,
						    449,
						    457,
						    461,
						    463,
						    467,
						    479,
						    487,
						    491,
						    499,
						    503,
						    509,
						    521,
						    523,
						    541,
						    547,
						    557,
						    563,
						    569,
						    571,
						    577,
						    587,
						    593,
						    599,
						    601,
						    607,
						    613,
						    617,
						    619,
						    631,
						    641,
						    643,
						    647,
						    653,
						    659,
						    661,
						    673,
						    677,
						    683,
						    691,
						    701,
						    709,
						    719,
						    727,
						    733,
						    739,
						    743,
						    751,
						    757,
						    761,
						    769,
						    773,
						    787,
						    797,
						    809,
						    811,
						    821,
						    823,
						    827,
						    829,
						    839,
						    853,
						    857,
						    859,
						    863,
						    877,
						    881,
						    883,
						    887,
						    907,
						    911,
						    919,
						    929,
						    937,
						    941,
						    947,
						    953,
						    967,
						    971,
						    977,
						    983,
						    991,
						    997,
						]

						if num in low_primes:
												return True

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

						return rabin_miller(UpperCamelCase_   )


def        _a  (     UpperCamelCase_					:				int = 1_024   ) ->   int:
						"""simple docstring"""







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


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






