

"""simple docstring"""


import random




def  _lowercase					(    __lowerCAmelCase							)       ->				Tuple:
      SCREAMING_SNAKE_CASE__   :      int      		=	num - 1
      SCREAMING_SNAKE_CASE__   :      List[Any]      		=	0

      while s % 2 == 0:
            SCREAMING_SNAKE_CASE__   :      Optional[int]      		=	s // 2
            t += 1

      for _ in range(5							):
            SCREAMING_SNAKE_CASE__   :      Dict      		=	random.randrange(2  ,					num - 1							)
            SCREAMING_SNAKE_CASE__   :      Any      		=	pow(UpperCAmelCase__  ,					UpperCAmelCase__  ,					UpperCAmelCase__							)
            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  _lowercase					(    __lowerCAmelCase							)       ->				Optional[int]:






      if num < 2:
            return False

      SCREAMING_SNAKE_CASE__   :      int      		=	[
          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  _lowercase					(    __lowerCAmelCase = 1024							)       ->				Any:






      while True:
            SCREAMING_SNAKE_CASE__   :      List[str]      		=	random.randrange(2 ** (keysize - 1)  ,					2 ** (keysize)							)
            if is_prime_low_num(UpperCAmelCase__							):
                  return num


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







"""simple docstring"""


from math import sqrt




def  _lowercase					(    __lowerCAmelCase							)       ->				bool:

      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(sqrt(__lowerCAmelCase							) + 1							)  ,					6							):
            if number % i == 0 or number % (i + 2) == 0:
                  return False
      return True




def  _lowercase					(    __lowerCAmelCase = 1_0001							)       ->				int:
      SCREAMING_SNAKE_CASE__   :      Dict      		=	0
      SCREAMING_SNAKE_CASE__   :      Tuple      		=	1
      while count != nth and number < 3:
            number += 1
            if is_prime(__lowerCAmelCase							):
                  count += 1
      while count != nth:
            number += 2
            if is_prime(__lowerCAmelCase							):
                  count += 1
      return number


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




