





import math




def 			lowerCAmelCase_			(				UpperCamelCase_	)  ->		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(__A	) + 1	)       ,		6	):
              if number % i == 0 or number % (i + 2) == 0:
                     return False
       return True




def 			lowerCAmelCase_			(				UpperCamelCase_ = 1000       ,		UpperCamelCase_ = 1000	)  ->		int:
       UpperCamelCase_ 							=	[0, 0, 0]  # length, a, b
       for a in range((a_limit * -1) + 1       ,		__A	):
              for b in range(2       ,		__A	):
                     if is_prime(__A	):
                            UpperCamelCase_ 							=	0
                            UpperCamelCase_ 							=	0
                            while is_prime((n**2) + (a * n) + b	):
                                   count += 1
                                   n += 1
                            if count > longest[0]:
                                   UpperCamelCase_ 							=	[count, a, b]
       UpperCamelCase_ 							=	longest[1] * longest[2]
       return ans


if __name__ == "__main__":
     print(solution(1_0_0_0, 1_0_0_0))





'''simple docstring'''







from __future__ import annotations

from collections.abc import Callable
from typing import Generic, TypeVar

a__    :   Dict								=			TypeVar('T')
a__    :   Optional[Any]								=			TypeVar('U')
class    lowercase_	(       Generic[T, U]  ):


    def __init__(	self    ,   a    ,   a       ):
          UpperCamelCase__      =							key
          UpperCamelCase__      =							val
          UpperCamelCase__      =							0
          UpperCamelCase__      =							None
          UpperCamelCase__      =							None


    def __repr__(	self       ):
          return "Node: key: {}, val: {}, freq: {}, has next: {}, has prev: {}".format(
              self.key    ,   self.val    ,   self.freq    ,   self.next is not None    ,   self.prev is not None       )

class    lowercase_	(       Generic[T, U]  ):


    def __init__(	self       ):
          UpperCamelCase__      =							DoubleLinkedListNode(a    ,   a       )
          UpperCamelCase__      =							DoubleLinkedListNode(a    ,   a       )
          UpperCamelCase__   ,      UpperCamelCase__      =							self.rear, self.head


    def __repr__(	self       ):
          UpperCamelCase__      =							["DoubleLinkedList"]
          UpperCamelCase__      =							self.head
          while node.next is not None:
                rep.append(str(a       )       )
                UpperCamelCase__      =							node.next
          rep.append(str(self.rear       )       )
          return ",\n    ".join(a       )


    def   __a				(	self    ,   a       ):
          UpperCamelCase__      =							self.rear.prev

          # All nodes other than self.head are guaranteed to have non-None previous
          assert previous is not None

          UpperCamelCase__      =							node
          UpperCamelCase__      =							previous
          UpperCamelCase__      =							node
          UpperCamelCase__      =							self.rear
          node.freq += 1
          self._position_node(a       )


    def   __a				(	self    ,   a       ):

          while node.prev is not None and node.prev.freq > node.freq:
                # swap node with previous node
                UpperCamelCase__      =							node.prev

                UpperCamelCase__      =							previous_node.prev
                UpperCamelCase__      =							node.prev
                UpperCamelCase__      =							previous_node
                UpperCamelCase__      =							node


    def   __a				(	self    ,   a       ):

          if node.prev is None or node.next is None:
                return None

          UpperCamelCase__      =							node.next
          UpperCamelCase__      =							node.prev
          UpperCamelCase__      =							None
          UpperCamelCase__      =							None
          return node

class    lowercase_	(       Generic[T, U]  ):
    __UpperCAmelCase       =      {}


    def __init__(	self    ,   a       ):
          UpperCamelCase__      =							DoubleLinkedList()
          UpperCamelCase__      =							capacity
          UpperCamelCase__      =							0
          UpperCamelCase__      =							0
          UpperCamelCase__      =							0
          UpperCamelCase__      =							{}


    def __repr__(	self       ):
          return (
              f'''CacheInfo(hits={self.hits}, misses={self.miss}, '''
              f'''capacity={self.capacity}, current_size={self.num_keys})'''
          )


    def __contains__(	self    ,   a       ):
          return key in self.cache


    def   __a				(	self    ,   a       ):

          if key in self.cache:
                self.hits += 1
                UpperCamelCase__      =							self.cache[key]
                UpperCamelCase__      =							self.list.remove(self.cache[key]       )
                assert node == value_node

                # node is guaranteed not None because it is in self.cache
                assert node is not None
                self.list.add(a       )
                return node.val
          self.miss += 1
          return None


    def   __a				(	self    ,   a    ,   a       ):

          if key not in self.cache:
                if self.num_keys >= self.capacity:
                      # delete first node when over capacity
                      UpperCamelCase__      =							self.list.head.next

                      # guaranteed to have a non-None first node when num_keys > 0
                      # explain to type checker via assertions
                      assert first_node is not None
                      assert first_node.key is not None
                      assert self.list.remove(a       ) is not None
                      # first_node guaranteed to be in list

                      del self.cache[first_node.key]
                      self.num_keys -= 1
                UpperCamelCase__      =							DoubleLinkedListNode(a    ,   a       )
                self.list.add(self.cache[key]       )
                self.num_keys += 1

          else:
                UpperCamelCase__      =							self.list.remove(self.cache[key]       )
                assert node is not None  # node guaranteed to be in list
                UpperCamelCase__      =							value
                self.list.add(a       )


    @classmethod
    def   __a				(	cls    ,   a = 1_28       ):

          def cache_decorator_inner(a       ) -> Callable[..., U]:
                def cache_decorator_wrapper(*a       ) -> U:
                      if func not in cls.decorator_function_to_instance_map:
                            UpperCamelCase__      =							LFUCache(a       )

                      UpperCamelCase__      =							cls.decorator_function_to_instance_map[func].get(args[0]       )
                      if result is None:
                            UpperCamelCase__      =							func(*a       )
                            cls.decorator_function_to_instance_map[func].put(args[0]    ,   a       )
                      return result

                def cache_info() -> LFUCache[T, U]:
                      return cls.decorator_function_to_instance_map[func]

                setattr(a    ,   "cache_info"    ,   a       )  # noqa: B010

                return cache_decorator_wrapper

          return cache_decorator_inner


if __name__ == "__main__":
  import doctest

  doctest.testmod()





