




def    A						(__A   :    int   ,				__A   :    bool = False       )   ->      bool:







     """simple docstring"""


     if n == 2:
          return True
     if not n % 2 or n < 2:
          return False
     if n > 5 and n % 10 not in (1, 3, 7, 9):  # can quickly check last digit
          return False
     if n > 3317044064679887385961981 and not allow_probable:
          raise ValueError(
              '''Warning: upper bound of deterministic test is exceeded. '''
              '''Pass allow_probable=True to allow probabilistic test. '''
              '''A return value of True indicates a probable prime.'''       )
     # array bounds provided by analysis
     UpperCAmelCase_          =							[
         2047,
         1373653,
         25326001,
         3215031751,
         2152302898747,
         3474749660383,
         341550071728321,
         1,
         3825123056546413051,
         1,
         1,
         318665857834031151167461,
         3317044064679887385961981,
     ]

     UpperCAmelCase_          =							[2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41]
     for idx, _p in enumerate(__A   ,				1       ):
          if n < _p:
               # then we have our last prime to check
               UpperCAmelCase_          =							primes[:idx]
               break
     UpperCAmelCase_     ,       UpperCAmelCase_          =							n - 1, 0
     # break up n -1 into a power of 2 (s) and
     # remaining odd component
     # essentially, solve for d * 2 ** s == n - 1
     while d % 2 == 0:
          d //= 2
          s += 1
     for prime in plist:
          UpperCAmelCase_          =							False
          for r in range(__A       ):
               UpperCAmelCase_          =							pow(__A   ,				d * 2**r   ,				__A       )
               # see article for analysis explanation for m
               if (r == 0 and m == 1) or ((m + 1) % n == 0):
                    UpperCAmelCase_          =							True
                    # this loop will not determine compositeness
                    break
          if pr:
               continue
          # if pr is False, then the above loop never evaluated to true,
          # and the n MUST be composite
          return False
     return True


def    A						()   ->      None:







     """simple docstring"""


     assert not miller_rabin(561       )
     assert miller_rabin(563       )
     # 2047

     assert not miller_rabin(838201       )
     assert miller_rabin(838207       )
     # 1_373_653

     assert not miller_rabin(17316001       )
     assert miller_rabin(17316017       )
     # 25_326_001

     assert not miller_rabin(3078386641       )
     assert miller_rabin(3078386653       )
     # 3_215_031_751

     assert not miller_rabin(1713045574801       )
     assert miller_rabin(1713045574819       )
     # 2_152_302_898_747

     assert not miller_rabin(2779799728307       )
     assert miller_rabin(2779799728327       )
     # 3_474_749_660_383

     assert not miller_rabin(113850023909441       )
     assert miller_rabin(113850023909527       )
     # 341_550_071_728_321

     assert not miller_rabin(1275041018848804351       )
     assert miller_rabin(1275041018848804391       )
     # 3_825_123_056_546_413_051

     assert not miller_rabin(79666464458507787791867       )
     assert miller_rabin(79666464458507787791951       )
     # 318_665_857_834_031_151_167_461

     assert not miller_rabin(552840677446647897660333       )
     assert miller_rabin(552840677446647897660359       )
     # 3_317_044_064_679_887_385_961_981
     # upper limit for probabilistic test


if __name__ == "__main__":
    test_miller_rabin()








'''simple docstring'''







def 				SCREAMING_SNAKE_CASE__	(   __A = 100	)       ->	int:
	_snake_case   						=		n * (n + 1) * (2 * n + 1) / 6
	_snake_case   						=		(n * (n + 1) / 2) ** 2
	return int(square_of_sum - sum_of_squares	)


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

