




import math

def 	lowerCAmelCase_	(     lowerCamelCase:				int  )    ->	bool:


      '''simple docstring'''




      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(lowerCamelCase  ) + 1  )   , 6  ):
            if number % i == 0 or number % (i + 2) == 0:
                  return False
      return True

def 	lowerCAmelCase_	(     lowerCamelCase:				int = 10_00   , lowerCamelCase:				int = 10_00  )    ->	int:


      '''simple docstring'''




      lowerCamelCase__					     = [0, 0, 0]  # length, a, b
      for a in range((a_limit * -1) + 1   , lowerCamelCase  ):
            for b in range(2   , lowerCamelCase  ):
                  if is_prime(lowerCamelCase  ):
                        lowerCamelCase__					     = 0
                        lowerCamelCase__					     = 0
                        while is_prime((n**2) + (a * n) + b  ):
                              count += 1
                              n += 1
                        if count > longest[0]:
                              lowerCamelCase__					     = [count, a, b]
      lowerCamelCase__					     = longest[1] * longest[2]
      return ans


if __name__ == "__main__":
      print(solution(1_000, 1_000))





from typing import List, Optional, Tuple, Union

import torch

from ...utils import randn_tensor
from ..pipeline_utils import DiffusionPipeline, ImagePipelineOutput






class 	A	(						_lowerCAmelCase						):
     """simple docstring"""

     def __init__(			self     :      Union[str, Any]		,     A__     :      Dict		,     A__     :      int	) ->				Union[str, Any]:
           super().__init__()
           self.register_modules(unet=A__		,     scheduler=A__	)




     @torch.no_grad()
     def __call__(			self     :      int		,     A__     :      int = 1		,     A__     :      Optional[Union[torch.Generator, List[torch.Generator]]] = None		,     A__     :      int = 1_0_0_0		,     A__     :      Optional[str] = "pil"		,     A__     :      bool = True		,     ) ->				Union[ImagePipelineOutput, Tuple]:
           # Sample gaussian noise to begin loop
           if isinstance(self.unet.config.sample_size		,     A__	):
                 lowerCamelCase__					     = (
                     batch_size,
                     self.unet.config.in_channels,
                     self.unet.config.sample_size,
                     self.unet.config.sample_size,
                 )
           else:
                 lowerCamelCase__					     = (batch_size, self.unet.config.in_channels, *self.unet.config.sample_size)

           if self.device.type == "mps":
                 # randn does not work reproducibly on mps
                 lowerCamelCase__					     = randn_tensor(A__		,     generator=A__	)
                 lowerCamelCase__					     = image.to(self.device	)
           else:
                 lowerCamelCase__					     = randn_tensor(A__		,     generator=A__		,     device=self.device	)

           # set step values
           self.scheduler.set_timesteps(A__	)

           for t in self.progress_bar(self.scheduler.timesteps	):
                 # 1. predict noise model_output
                 lowerCamelCase__					     = self.unet(A__		,     A__	).sample

                 # 2. compute previous image: x_t -> x_t-1
                 lowerCamelCase__					     = self.scheduler.step(A__		,     A__		,     A__		,     generator=A__	).prev_sample

           lowerCamelCase__					     = (image / 2 + 0.5).clamp(0		,     1	)
           lowerCamelCase__					     = image.cpu().permute(0		,     2		,     3		,     1	).numpy()
           if output_type == "pil":
                 lowerCamelCase__					     = self.numpy_to_pil(A__	)

           if not return_dict:
                 return (image,)

           return ImagePipelineOutput(images=A__	)