
def 							a		(				snake_case__:   int		,  snake_case__:   Optional[Any] = False   ):


				'''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_317_044_064_679_887_385_961_981 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
				lowercase_	    =   [
				    2_047,
				    1_373_653,
				    25_326_001,
				    3_215_031_751,
				    2_152_302_898_747,
				    3_474_749_660_383,
				    341_550_071_728_321,
				    1,
				    3_825_123_056_546_413_051,
				    1,
				    1,
				    318_665_857_834_031_151_167_461,
				    3_317_044_064_679_887_385_961_981,
				]

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


				'''simple docstring'''
				assert not miller_rabin(561   )
				assert miller_rabin(563   )
				# 2047

				assert not miller_rabin(838_201   )
				assert miller_rabin(838_207   )
				# 1_373_653

				assert not miller_rabin(17_316_001   )
				assert miller_rabin(17_316_017   )
				# 25_326_001

				assert not miller_rabin(3_078_386_641   )
				assert miller_rabin(3_078_386_653   )
				# 3_215_031_751

				assert not miller_rabin(1_713_045_574_801   )
				assert miller_rabin(1_713_045_574_819   )
				# 2_152_302_898_747

				assert not miller_rabin(2_779_799_728_307   )
				assert miller_rabin(2_779_799_728_327   )
				# 3_474_749_660_383

				assert not miller_rabin(113_850_023_909_441   )
				assert miller_rabin(113_850_023_909_527   )
				# 341_550_071_728_321

				assert not miller_rabin(1_275_041_018_848_804_351   )
				assert miller_rabin(1_275_041_018_848_804_391   )
				# 3_825_123_056_546_413_051

				assert not miller_rabin(79_666_464_458_507_787_791_867   )
				assert miller_rabin(79_666_464_458_507_787_791_951   )
				# 318_665_857_834_031_151_167_461

				assert not miller_rabin(552_840_677_446_647_897_660_333   )
				assert miller_rabin(552_840_677_446_647_897_660_359   )
				# 3_317_044_064_679_887_385_961_981
				# upper limit for probabilistic test


if __name__ == "__main__":
						test_miller_rabin()










from typing import TYPE_CHECKING

from ...utils import (
    OptionalDependencyNotAvailable,
    _LazyModule,
    is_torch_available,
)


__a							     =					{
    'configuration_encodec': [
        'ENCODEC_PRETRAINED_CONFIG_ARCHIVE_MAP',
        'EncodecConfig',
    ],
    'feature_extraction_encodec': ['EncodecFeatureExtractor'],
}

try:
     if not is_torch_available():
          raise OptionalDependencyNotAvailable()
except OptionalDependencyNotAvailable:
     pass
else:
     __a							     =					[
         'ENCODEC_PRETRAINED_MODEL_ARCHIVE_LIST',
         'EncodecModel',
         'EncodecPreTrainedModel',
     ]

if TYPE_CHECKING:
     from .configuration_encodec import (
         ENCODEC_PRETRAINED_CONFIG_ARCHIVE_MAP,
         EncodecConfig,
     )
     from .feature_extraction_encodec import EncodecFeatureExtractor

     try:
          if not is_torch_available():
               raise OptionalDependencyNotAvailable()
     except OptionalDependencyNotAvailable:
          pass
     else:
          from .modeling_encodec import (
              ENCODEC_PRETRAINED_MODEL_ARCHIVE_LIST,
              EncodecModel,
              EncodecPreTrainedModel,
          )

else:
     import sys

     __a							     =					_LazyModule(__name__, globals()['__file__'], _import_structure, module_spec=__spec__)