


import random

def 						_UpperCAmelCase				(			SCREAMING_SNAKE_CASE__			:					int				):
     __UpperCamelCase    							=num - 1
     __UpperCamelCase    							=0

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

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

def 						_UpperCAmelCase				(			SCREAMING_SNAKE_CASE__			:					int				):
     if num < 2:
          return False

     __UpperCamelCase    							=[
         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(SCREAMING_SNAKE_CASE__				)

def 						_UpperCAmelCase				(			SCREAMING_SNAKE_CASE__			:					int = 10_24				):
     while True:
          __UpperCamelCase    							=random.randrange(2 ** (keysize - 1)			,     2 ** (keysize)				)
          if is_prime_low_num(SCREAMING_SNAKE_CASE__				):
               return num


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









from numpy import exp, pi, sqrt

def 						_UpperCAmelCase				(			SCREAMING_SNAKE_CASE__			:					Tuple			,     SCREAMING_SNAKE_CASE__			:					float = 0.0			,     SCREAMING_SNAKE_CASE__			:					float = 1.0				):
     return 1 / sqrt(2 * pi * sigma**2				) * exp(-((x - mu) ** 2) / (2 * sigma**2)				)


if __name__ == "__main__":
   import doctest

   doctest.testmod()





