
import math




def 							__SCREAMING_SNAKE_CASE       (		SCREAMING_SNAKE_CASE	:				List[Any]      )					->						str:

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




def 							__SCREAMING_SNAKE_CASE       (		SCREAMING_SNAKE_CASE	:				str = 1000   ,							SCREAMING_SNAKE_CASE	:				int = 1000      )					->						Optional[int]:
				__lowercase								= [0, 0, 0]  # length, a, b
				for a in range((a_limit * -1) + 1   ,							SCREAMING_SNAKE_CASE_      ):
								for b in range(2   ,							SCREAMING_SNAKE_CASE_      ):
												if is_prime(SCREAMING_SNAKE_CASE_      ):
																__lowercase								= 0
																__lowercase								= 0
																while is_prime((n**2) + (a * n) + b      ):
																				count += 1
																				n += 1
																if count > longest[0]:
																				__lowercase								= [count, a, b]
				__lowercase								= longest[1] * longest[2]
				return ans


if __name__ == "__main__":
							print(solution(1000, 1000))







from typing import List, Optional, Union

import numpy as np
import PIL
import torch
from PIL import Image

from ...models import UNetaDConditionModel, VQModel
from ...pipelines import DiffusionPipeline
from ...pipelines.pipeline_utils import ImagePipelineOutput
from ...schedulers import DDPMScheduler
from ...utils import (
    is_accelerate_available,
    is_accelerate_version,
    logging,
    randn_tensor,
    replace_example_docstring,
)


SCREAMING_SNAKE_CASE__							=    logging.get_logger(__name__)  # pylint: disable=invalid-name

SCREAMING_SNAKE_CASE__							=    """
    Examples:
        ```py
        >>> import torch
        >>> import numpy as np

        >>> from diffusers import KandinskyV22PriorEmb2EmbPipeline, KandinskyV22ControlnetImg2ImgPipeline
        >>> from transformers import pipeline
        >>> from diffusers.utils import load_image


        >>> def make_hint(image, depth_estimator):
        ...     image = depth_estimator(image)[\"depth\"]
        ...     image = np.array(image)
        ...     image = image[:, :, None]
        ...     image = np.concatenate([image, image, image], axis=2)
        ...     detected_map = torch.from_numpy(image).float() / 255.0
        ...     hint = detected_map.permute(2, 0, 1)
        ...     return hint


        >>> depth_estimator = pipeline(\"depth-estimation\")

        >>> pipe_prior = KandinskyV22PriorEmb2EmbPipeline.from_pretrained(
        ...     \"kandinsky-community/kandinsky-2-2-prior\", torch_dtype=torch.float16
        ... )
        >>> pipe_prior = pipe_prior.to(\"cuda\")

        >>> pipe = KandinskyV22ControlnetImg2ImgPipeline.from_pretrained(
        ...     \"kandinsky-community/kandinsky-2-2-controlnet-depth\", torch_dtype=torch.float16
        ... )
        >>> pipe = pipe.to(\"cuda\")

        >>> img = load_image(
        ...     \"https://huggingface.co/datasets/hf-internal-testing/diffusers-images/resolve/main\"
        ...     \"/kandinsky/cat.png\"
        ... ).resize((768, 768))


        >>> hint = make_hint(img, depth_estimator).unsqueeze(0).half().to(\"cuda\")

        >>> prompt = \"A robot, 4k photo\"
        >>> negative_prior_prompt = \"lowres, text, error, cropped, worst quality, low quality, jpeg artifacts, ugly, duplicate, morbid, mutilated, out of frame, extra fingers, mutated hands, poorly drawn hands, poorly drawn face, mutation, deformed, blurry, dehydrated, bad anatomy, bad proportions, extra limbs, cloned face, disfigured, gross proportions, malformed limbs, missing arms, missing legs, extra arms, extra legs, fused fingers, too many fingers, long neck, username, watermark, signature\"

        >>> generator = torch.Generator(device=\"cuda\").manual_seed(43)

        >>> img_emb = pipe_prior(prompt=prompt, image=img, strength=0.85, generator=generator)
        >>> negative_emb = pipe_prior(prompt=negative_prior_prompt, image=img, strength=1, generator=generator)

        >>> images = pipe(
        ...     image=img,
        ...     strength=0.5,
        ...     image_embeds=img_emb.image_embeds,
        ...     negative_image_embeds=negative_emb.image_embeds,
        ...     hint=hint,
        ...     num_inference_steps=50,
        ...     generator=generator,
        ...     height=768,
        ...     width=768,
        ... ).images

        >>> images[0].save(\"robot_cat.png\")
        ```
"""




def 							__SCREAMING_SNAKE_CASE       (		SCREAMING_SNAKE_CASE	:				List[Any]   ,							SCREAMING_SNAKE_CASE	:				Dict   ,							SCREAMING_SNAKE_CASE	:				Any=8      )					->						Any:
				__lowercase								= height // scale_factor**2
				if height % scale_factor**2 != 0:
								new_height += 1
				__lowercase								= width // scale_factor**2
				if width % scale_factor**2 != 0:
								new_width += 1
				return new_height * scale_factor, new_width * scale_factor




def 							__SCREAMING_SNAKE_CASE       (		SCREAMING_SNAKE_CASE	:				List[Any]   ,							SCREAMING_SNAKE_CASE	:				Any=512   ,							SCREAMING_SNAKE_CASE	:				List[str]=512      )					->						str:
				__lowercase								= pil_image.resize((w, h)   ,							resample=Image.BICUBIC   ,							reducing_gap=1      )
				__lowercase								= np.array(pil_image.convert('RGB'      )      )
				__lowercase								= arr.astype(np.floataa      ) / 127.5 - 1
				__lowercase								= np.transpose(SCREAMING_SNAKE_CASE   ,							[2, 0, 1]      )
				__lowercase								= torch.from_numpy(SCREAMING_SNAKE_CASE      ).unsqueeze(0      )
				return image
class      A__       (						lowerCAmelCase__    ):





							def __init__(			self  :       str					,			_UpperCAmelCase  :       UNetaDConditionModel					,			_UpperCAmelCase  :       DDPMScheduler					,			_UpperCAmelCase  :       VQModel					,			)						->		Optional[Any]:



											"""simple docstring"""


											super().__init__()

											self.register_modules(
											    unet=_UpperCAmelCase					,			scheduler=_UpperCAmelCase					,			movq=_UpperCAmelCase					,			)
											__lowercase								= 2 ** (len(self.movq.config.block_out_channels    ) - 1)





							def       a__			(			self  :       List[str]					,			_UpperCAmelCase  :       Dict					,			_UpperCAmelCase  :       int					,			_UpperCAmelCase  :       Dict    )						->		List[str]:



											"""simple docstring"""


											__lowercase								= min(int(num_inference_steps * strength    )					,			_UpperCAmelCase    )

											__lowercase								= max(num_inference_steps - init_timestep					,			0    )
											__lowercase								= self.scheduler.timesteps[t_start:]

											return timesteps, num_inference_steps - t_start





							def       a__			(			self  :       List[str]					,			_UpperCAmelCase  :       int					,			_UpperCAmelCase  :       List[str]					,			_UpperCAmelCase  :       Any					,			_UpperCAmelCase  :       Optional[int]					,			_UpperCAmelCase  :       Optional[Any]					,			_UpperCAmelCase  :       List[str]					,			_UpperCAmelCase  :       List[Any]=None    )						->		Optional[Any]:



											"""simple docstring"""


											if not isinstance(_UpperCAmelCase					,			(torch.Tensor, PIL.Image.Image, list)    ):
															raise ValueError(
															    f"""`image` has to be of type `torch.Tensor`, `PIL.Image.Image` or list but is {type(_UpperCAmelCase    )}"""    )

											__lowercase								= image.to(device=_UpperCAmelCase					,			dtype=_UpperCAmelCase    )

											__lowercase								= batch_size * num_images_per_prompt

											if image.shape[1] == 4:
															__lowercase								= image

											else:
															if isinstance(_UpperCAmelCase					,			_UpperCAmelCase    ) and len(_UpperCAmelCase    ) != batch_size:
																			raise ValueError(
																			    f"""You have passed a list of generators of length {len(_UpperCAmelCase    )}, but requested an effective batch"""
																			    f""" size of {batch_size}. Make sure the batch size matches the length of the generators."""    )

															elif isinstance(_UpperCAmelCase					,			_UpperCAmelCase    ):
																			__lowercase								= [
																			    self.movq.encode(image[i : i + 1]    ).latent_dist.sample(generator[i]    ) for i in range(_UpperCAmelCase    )
																			]
																			__lowercase								= torch.cat(_UpperCAmelCase					,			dim=0    )
															else:
																			__lowercase								= self.movq.encode(_UpperCAmelCase    ).latent_dist.sample(_UpperCAmelCase    )

															__lowercase								= self.movq.config.scaling_factor * init_latents

											__lowercase								= torch.cat([init_latents]					,			dim=0    )

											__lowercase								= init_latents.shape
											__lowercase								= randn_tensor(_UpperCAmelCase					,			generator=_UpperCAmelCase					,			device=_UpperCAmelCase					,			dtype=_UpperCAmelCase    )

											# get latents
											__lowercase								= self.scheduler.add_noise(_UpperCAmelCase					,			_UpperCAmelCase					,			_UpperCAmelCase    )

											__lowercase								= init_latents

											return latents





							def       a__			(			self  :       List[Any]					,			_UpperCAmelCase  :       Dict=0    )						->		List[str]:



											"""simple docstring"""


											if is_accelerate_available():
															from accelerate import cpu_offload
											else:
															raise ImportError('Please install accelerate via `pip install accelerate`'    )

											__lowercase								= torch.device(f"""cuda:{gpu_id}"""    )

											__lowercase								= [
											    self.unet,
											    self.movq,
											]
											for cpu_offloaded_model in models:
															if cpu_offloaded_model is not None:
																			cpu_offload(_UpperCAmelCase					,			_UpperCAmelCase    )





							def       a__			(			self  :       str					,			_UpperCAmelCase  :       Tuple=0    )						->		List[str]:



											"""simple docstring"""


											if is_accelerate_available() and is_accelerate_version('>='					,			'0.17.0.dev0'    ):
															from accelerate import cpu_offload_with_hook
											else:
															raise ImportError('`enable_model_cpu_offload` requires `accelerate v0.17.0` or higher.'    )

											__lowercase								= torch.device(f"""cuda:{gpu_id}"""    )

											if self.device.type != "cpu":
															self.to('cpu'					,			silence_dtype_warnings=_UpperCAmelCase    )
															torch.cuda.empty_cache()  # otherwise we don't see the memory savings (but they probably exist)

											__lowercase								= None
											for cpu_offloaded_model in [self.unet, self.movq]:
															__lowercase		, __lowercase								= cpu_offload_with_hook(_UpperCAmelCase					,			_UpperCAmelCase					,			prev_module_hook=_UpperCAmelCase    )

											# We'll offload the last model manually.
											__lowercase								= hook





							@property
							# Copied from diffusers.pipelines.stable_diffusion.pipeline_stable_diffusion.StableDiffusionPipeline._execution_device
							def       a__			(			self  :       int    )						->		Optional[int]:



											"""simple docstring"""


											if not hasattr(self.unet					,			'_hf_hook'    ):
															return self.device
											for module in self.unet.modules():
															if (
															    hasattr(_UpperCAmelCase					,			'_hf_hook'    )
															    and hasattr(module._hf_hook					,			'execution_device'    )
															    and module._hf_hook.execution_device is not None
															):
																			return torch.device(module._hf_hook.execution_device    )
											return self.device





							@torch.no_grad()
							@replace_example_docstring(_UpperCAmelCase    )
							def __call__(			self  :       Optional[Any]					,			_UpperCAmelCase  :       Union[torch.FloatTensor, List[torch.FloatTensor]]					,			_UpperCAmelCase  :       Union[torch.FloatTensor, PIL.Image.Image, List[torch.FloatTensor], List[PIL.Image.Image]]					,			_UpperCAmelCase  :       Union[torch.FloatTensor, List[torch.FloatTensor]]					,			_UpperCAmelCase  :       torch.FloatTensor					,			_UpperCAmelCase  :       int = 5_12					,			_UpperCAmelCase  :       int = 5_12					,			_UpperCAmelCase  :       int = 1_00					,			_UpperCAmelCase  :       float = 4.0					,			_UpperCAmelCase  :       float = 0.3					,			_UpperCAmelCase  :       int = 1					,			_UpperCAmelCase  :       Optional[Union[torch.Generator, List[torch.Generator]]] = None					,			_UpperCAmelCase  :       Optional[str] = "pil"					,			_UpperCAmelCase  :       bool = True					,			)						->		Union[str, Any]:



											"""simple docstring"""


											__lowercase								= self._execution_device

											__lowercase								= guidance_scale > 1.0

											if isinstance(_UpperCAmelCase					,			_UpperCAmelCase    ):
															__lowercase								= torch.cat(_UpperCAmelCase					,			dim=0    )
											if isinstance(_UpperCAmelCase					,			_UpperCAmelCase    ):
															__lowercase								= torch.cat(_UpperCAmelCase					,			dim=0    )
											if isinstance(_UpperCAmelCase					,			_UpperCAmelCase    ):
															__lowercase								= torch.cat(_UpperCAmelCase					,			dim=0    )

											__lowercase								= image_embeds.shape[0]

											if do_classifier_free_guidance:
															__lowercase								= image_embeds.repeat_interleave(_UpperCAmelCase					,			dim=0    )
															__lowercase								= negative_image_embeds.repeat_interleave(_UpperCAmelCase					,			dim=0    )
															__lowercase								= hint.repeat_interleave(_UpperCAmelCase					,			dim=0    )

											__lowercase								= torch.cat([negative_image_embeds, image_embeds]					,			dim=0    ).to(dtype=self.unet.dtype					,			device=_UpperCAmelCase    )
											__lowercase								= torch.cat([hint, hint]					,			dim=0    ).to(dtype=self.unet.dtype					,			device=_UpperCAmelCase    )

											if not isinstance(_UpperCAmelCase					,			_UpperCAmelCase    ):
															__lowercase								= [image]
											if not all(isinstance(_UpperCAmelCase					,			(PIL.Image.Image, torch.Tensor)    ) for i in image    ):
															raise ValueError(
															    f"""Input is in incorrect format: {[type(_UpperCAmelCase    ) for i in image]}. Currently, we only support  PIL image and pytorch tensor"""    )

											__lowercase								= torch.cat([prepare_image(_UpperCAmelCase					,			_UpperCAmelCase					,			_UpperCAmelCase    ) for i in image]					,			dim=0    )
											__lowercase								= image.to(dtype=image_embeds.dtype					,			device=_UpperCAmelCase    )

											__lowercase								= self.movq.encode(_UpperCAmelCase    )['latents']
											__lowercase								= latents.repeat_interleave(_UpperCAmelCase					,			dim=0    )
											self.scheduler.set_timesteps(_UpperCAmelCase					,			device=_UpperCAmelCase    )
											__lowercase		, __lowercase								= self.get_timesteps(_UpperCAmelCase					,			_UpperCAmelCase					,			_UpperCAmelCase    )
											__lowercase								= timesteps[:1].repeat(batch_size * num_images_per_prompt    )
											__lowercase		, __lowercase								= downscale_height_and_width(_UpperCAmelCase					,			_UpperCAmelCase					,			self.movq_scale_factor    )
											__lowercase								= self.prepare_latents(
											    _UpperCAmelCase					,			_UpperCAmelCase					,			_UpperCAmelCase					,			_UpperCAmelCase					,			image_embeds.dtype					,			_UpperCAmelCase					,			_UpperCAmelCase    )
											for i, t in enumerate(self.progress_bar(_UpperCAmelCase    )    ):
															# expand the latents if we are doing classifier free guidance
															__lowercase								= torch.cat([latents] * 2    ) if do_classifier_free_guidance else latents

															__lowercase								= {'image_embeds': image_embeds, 'hint': hint}
															__lowercase								= self.unet(
															    sample=_UpperCAmelCase					,			timestep=_UpperCAmelCase					,			encoder_hidden_states=_UpperCAmelCase					,			added_cond_kwargs=_UpperCAmelCase					,			return_dict=_UpperCAmelCase					,			)[0]

															if do_classifier_free_guidance:
																			__lowercase		, __lowercase								= noise_pred.split(latents.shape[1]					,			dim=1    )
																			__lowercase		, __lowercase								= noise_pred.chunk(2    )
																			__lowercase		, __lowercase								= variance_pred.chunk(2    )
																			__lowercase								= noise_pred_uncond + guidance_scale * (noise_pred_text - noise_pred_uncond)
																			__lowercase								= torch.cat([noise_pred, variance_pred_text]					,			dim=1    )

															if not (
															    hasattr(self.scheduler.config					,			'variance_type'    )
															    and self.scheduler.config.variance_type in ["learned", "learned_range"]
															):
																			__lowercase		, __lowercase								= noise_pred.split(latents.shape[1]					,			dim=1    )

															# compute the previous noisy sample x_t -> x_t-1

															__lowercase								= self.scheduler.step(
															    _UpperCAmelCase					,			_UpperCAmelCase					,			_UpperCAmelCase					,			generator=_UpperCAmelCase					,			)[0]

											# post-processing
											__lowercase								= self.movq.decode(_UpperCAmelCase					,			force_not_quantize=_UpperCAmelCase    )['sample']

											if output_type not in ["pt", "np", "pil"]:
															raise ValueError(f"""Only the output types `pt`, `pil` and `np` are supported not output_type={output_type}"""    )

											if output_type in ["np", "pil"]:
															__lowercase								= image * 0.5 + 0.5
															__lowercase								= image.clamp(0					,			1    )
															__lowercase								= image.cpu().permute(0					,			2					,			3					,			1    ).float().numpy()

											if output_type == "pil":
															__lowercase								= self.numpy_to_pil(_UpperCAmelCase    )

											if not return_dict:
															return (image,)

											return ImagePipelineOutput(images=_UpperCAmelCase    )





