

'''simple docstring'''



UpperCAmelCase :       Dict			   =     {
    '''Pillow''': '''Pillow''',
    '''accelerate''': '''accelerate>=0.11.0''',
    '''compel''': '''compel==0.1.8''',
    '''black''': '''black~=23.1''',
    '''datasets''': '''datasets''',
    '''filelock''': '''filelock''',
    '''flax''': '''flax>=0.4.1''',
    '''hf-doc-builder''': '''hf-doc-builder>=0.3.0''',
    '''huggingface-hub''': '''huggingface-hub>=0.13.2''',
    '''requests-mock''': '''requests-mock==1.10.0''',
    '''importlib_metadata''': '''importlib_metadata''',
    '''invisible-watermark''': '''invisible-watermark''',
    '''isort''': '''isort>=5.5.4''',
    '''jax''': '''jax>=0.2.8,!=0.3.2''',
    '''jaxlib''': '''jaxlib>=0.1.65''',
    '''Jinja2''': '''Jinja2''',
    '''k-diffusion''': '''k-diffusion>=0.0.12''',
    '''torchsde''': '''torchsde''',
    '''note_seq''': '''note_seq''',
    '''librosa''': '''librosa''',
    '''numpy''': '''numpy''',
    '''omegaconf''': '''omegaconf''',
    '''parameterized''': '''parameterized''',
    '''protobuf''': '''protobuf>=3.20.3,<4''',
    '''pytest''': '''pytest''',
    '''pytest-timeout''': '''pytest-timeout''',
    '''pytest-xdist''': '''pytest-xdist''',
    '''ruff''': '''ruff>=0.0.241''',
    '''safetensors''': '''safetensors''',
    '''sentencepiece''': '''sentencepiece>=0.1.91,!=0.1.92''',
    '''scipy''': '''scipy''',
    '''onnx''': '''onnx''',
    '''regex''': '''regex!=2019.12.17''',
    '''requests''': '''requests''',
    '''tensorboard''': '''tensorboard''',
    '''torch''': '''torch>=1.4''',
    '''torchvision''': '''torchvision''',
    '''transformers''': '''transformers>=4.25.1''',
    '''urllib3''': '''urllib3<=2.0.0''',
}













'''simple docstring'''




import math






def        _lowerCAmelCase			(  __snake_case  : int			)     ->     bool:

     if 1 < number < 4:
          # 2 and 3 are primes
          return True
     elif number < 2 or number % 2 == 0 or number % 3 == 0:
          # Negatives, 0, 1, all even numbers, all multiples of 3 are not primes
          return False

     # All primes number are in format of 6k +/- 1
     for i in range(5 ,							int(math.sqrt(__snake_case			) + 1			) ,							6			):
          if number % i == 0 or number % (i + 2) == 0:
               return False
     return True






def        _lowerCAmelCase			(  __snake_case  : float = 0.1			)     ->     int:
     __A						: Tuple      =       3
     __A						: Optional[int]      =       3

     while primes / (2 * j - 1) >= ratio:
          for i in range(j * j + j + 1 ,							(j + 2) * (j + 2) ,							j + 1			):
               primes += is_prime(__snake_case			)
          j += 2
     return j


if __name__ == "__main__":
  import doctest

  doctest.testmod()