





'''simple docstring'''






def 					_a						(       _lowerCamelCase    ,	_lowerCamelCase = 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 > 3_3170_4406_4679_8873_8596_1981 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
			__snake_case					:      List[Any]							      =     [
			    2047,
			    137_3653,
			    2532_6001,
			    32_1503_1751,
			    2_1523_0289_8747,
			    3_4747_4966_0383,
			    341_5500_7172_8321,
			    1,
			    382_5123_0565_4641_3051,
			    1,
			    1,
			    3186_6585_7834_0311_5116_7461,
			    3_3170_4406_4679_8873_8596_1981,
			]

			__snake_case					:      str							      =     [2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41]
			for idx, _p in enumerate(_lowerCamelCase    ,	1 ):
						if n < _p:
									# then we have our last prime to check
									__snake_case					:      Tuple							      =     primes[:idx]
									break
			__snake_case      ,     __snake_case					:      Union[str, 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:
						__snake_case					:      Tuple							      =     False
						for r in range(_lowerCamelCase ):
									__snake_case					:      Tuple							      =     pow(_lowerCamelCase    ,	d * 2**r    ,	_lowerCamelCase )
									# see article for analysis explanation for m
									if (r == 0 and m == 1) or ((m + 1) % n == 0):
												__snake_case					:      List[str]							      =     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(83_8201 )
			assert miller_rabin(83_8207 )
			# 1_373_653

			assert not miller_rabin(1731_6001 )
			assert miller_rabin(1731_6017 )
			# 25_326_001

			assert not miller_rabin(30_7838_6641 )
			assert miller_rabin(30_7838_6653 )
			# 3_215_031_751

			assert not miller_rabin(1_7130_4557_4801 )
			assert miller_rabin(1_7130_4557_4819 )
			# 2_152_302_898_747

			assert not miller_rabin(2_7797_9972_8307 )
			assert miller_rabin(2_7797_9972_8327 )
			# 3_474_749_660_383

			assert not miller_rabin(113_8500_2390_9441 )
			assert miller_rabin(113_8500_2390_9527 )
			# 341_550_071_728_321

			assert not miller_rabin(127_5041_0188_4880_4351 )
			assert miller_rabin(127_5041_0188_4880_4391 )
			# 3_825_123_056_546_413_051

			assert not miller_rabin(796_6646_4458_5077_8779_1867 )
			assert miller_rabin(796_6646_4458_5077_8779_1951 )
			# 318_665_857_834_031_151_167_461

			assert not miller_rabin(5528_4067_7446_6478_9766_0333 )
			assert miller_rabin(5528_4067_7446_6478_9766_0359 )
			# 3_317_044_064_679_887_385_961_981
			# upper limit for probabilistic test


if __name__ == "__main__":
	test_miller_rabin()










from __future__ import annotations


def 				__lowerCamelCase     (							__lowerCAmelCase							:					str					,							__lowerCAmelCase							:					Tuple					,							__lowerCAmelCase							:					int					,							__lowerCAmelCase							:					Tuple							)     ->  Optional[Any]:  # noqa: E741
   while r - l > 1:
      __UpperCamelCase	:			int					=					(l + r) // 2
      if v[m] >= key:
         __UpperCamelCase	:			Dict					=					m
      else:
         __UpperCamelCase	:			Optional[Any]					=					m  # noqa: E741
   return r


def 				__lowerCamelCase     (							__lowerCAmelCase							:					list[int]							)     ->  int:
   if len(__lowerCAmelCase							) == 0:
      return 0

   __UpperCamelCase	:			Optional[Any]					=					[0] * len(__lowerCAmelCase							)
   __UpperCamelCase	:			Optional[int]					=					1

   __UpperCamelCase	:			Union[str, Any]					=					v[0]

   for i in range(1					,							len(__lowerCAmelCase							)							):
      if v[i] < tail[0]:
         __UpperCamelCase	:			Dict					=					v[i]
      elif v[i] > tail[length - 1]:
         __UpperCamelCase	:			Optional[int]					=					v[i]
         length += 1
      else:
         __UpperCamelCase	:			List[Any]					=					v[i]

   return length


if __name__ == "__main__":
       import doctest

       doctest.testmod()



