


def     __lowerCAmelCase						(							snake_case	:    int						,      snake_case	:    bool = False )       ->							bool:
   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
   __lowerCamelCase:				List[Any]     =	[
       2047,
       1373653,
       25326001,
       3215031751,
       2152302898747,
       3474749660383,
       341550071728321,
       1,
       3825123056546413051,
       1,
       1,
       318665857834031151167461,
       3317044064679887385961981,
   ]

   __lowerCamelCase:				List[str]     =	[2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41]
   for idx, _p in enumerate(snake_case						,      1 ):
      if n < _p:
         # then we have our last prime to check
         __lowerCamelCase:				Optional[int]     =	primes[:idx]
         break
   __lowerCamelCase   ,     __lowerCamelCase:				List[Any]     =	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:
      __lowerCamelCase:				List[Any]     =	False
      for r in range(snake_case ):
         __lowerCamelCase:				Dict     =	pow(snake_case						,      d * 2**r						,      snake_case )
         # see article for analysis explanation for m
         if (r == 0 and m == 1) or ((m + 1) % n == 0):
            __lowerCamelCase:				Optional[Any]     =	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     __lowerCAmelCase						(							)       ->							None:
   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()






def     __lowerCAmelCase						(							snake_case	:    int = 100 )       ->							int:
   __lowerCamelCase:				List[Any]     =	0
   __lowerCamelCase:				int     =	0
   for i in range(1						,      n + 1 ):
      sum_of_squares += i**2
      sum_of_ints += i
   return sum_of_ints**2 - sum_of_squares


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


