





def 	_a						(   UpperCAmelCase      ,  UpperCAmelCase = 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
   lowerCamelCase__       :						Optional[int]        =							[
       2047,
       1373653,
       25326001,
       3215031751,
       2152302898747,
       3474749660383,
       341550071728321,
       1,
       3825123056546413051,
       1,
       1,
       318665857834031151167461,
       3317044064679887385961981,
   ]

   lowerCamelCase__       :						Union[str, Any]        =							[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__       :						List[str]        =							primes[:idx]
         break
   lowerCamelCase__       ,							lowerCamelCase__       :						List[str]        =							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__       :						Dict        =							False
      for r in range(__snake_case					):
         lowerCamelCase__       :						str        =							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__       :						int        =							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()










from math import ceil, sqrt



def 	_a						(   UpperCAmelCase = 1000000					)    ->    int:







   """simple docstring"""
   lowerCamelCase__       :						Any        =							0

   for outer_width in range(3      ,  (limit // 4) + 2					):
      if outer_width**2 > limit:
         lowerCamelCase__       :						List[Any]        =							max(ceil(sqrt(outer_width**2 - limit					)					)      ,  1					)
      else:
         lowerCamelCase__       :						Union[str, Any]        =							1
      if (outer_width - hole_width_lower_bound) % 2:
         hole_width_lower_bound += 1

      answer += (outer_width - hole_width_lower_bound - 2) // 2 + 1

   return answer


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



