code: 
  
  
  
  """simple docstring"""
  
  
  
  
  import math
  
  def 		__snake_case		(   SCREAMING_SNAKE_CASE__			:		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(SCREAMING_SNAKE_CASE__       ) + 1       )		,						6       ):
            if number % i == 0 or number % (i + 2) == 0:
                 return False
       return True
  
  def 		__snake_case		(   SCREAMING_SNAKE_CASE__			:		int = 1_000		,						SCREAMING_SNAKE_CASE__			:		int = 1_000       )						->     int:
  
  
  
  
  
       '''simple docstring'''
  
       _UpperCAmelCase    :			List[Any]              =					[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__       ):
                      _UpperCAmelCase    :			int              =					0
                      _UpperCAmelCase    :			Optional[Any]              =					0
                      while is_prime((n**2) + (a * n) + b       ):
                           count += 1
                           n += 1
                      if count > longest[0]:
                           _UpperCAmelCase    :			Union[str, Any]              =					[count, a, b]
       _UpperCAmelCase    :			Any              =					longest[1] * longest[2]
       return ans
  
  
  if __name__ == "__main__":
   print(solution(10_00, 10_00))
  
style_context: 
  
  
  
  """simple docstring"""
  
  
  
  
  _lowerCAmelCase       :      str							=							{
      "0": 1,
      "1": 1,
      "2": 2,
      "3": 6,
      "4": 24,
      "5": 1_20,
      "6": 7_20,
      "7": 50_40,
      "8": 4_03_20,
      "9": 36_28_80,
  }
  
  _lowerCAmelCase       :      Optional[int]							=							{1_45: 1_45}
  
  _lowerCAmelCase       :      int							=							{
      1_45: 0,
      1_69: 3,
      3_63_01: 3,
      14_54: 3,
      8_71: 2,
      4_53_61: 2,
      8_72: 2,
  }
  
  def 		__snake_case		(   SCREAMING_SNAKE_CASE__			:		int       )						->     int:
  
  
  
  
  
       '''simple docstring'''
  
       if n in CACHE_SUM_DIGIT_FACTORIALS:
            return CACHE_SUM_DIGIT_FACTORIALS[n]
       _UpperCAmelCase    :			Dict              =					sum(DIGIT_FACTORIALS[let] for let in str(SCREAMING_SNAKE_CASE__       )       )
       _UpperCAmelCase    :			Any              =					ret
       return ret
  
  def 		__snake_case		(   SCREAMING_SNAKE_CASE__			:		int		,						SCREAMING_SNAKE_CASE__			:		set | None = None       )						->     int:
  
  
  
  
  
       '''simple docstring'''
  
       _UpperCAmelCase    :			Dict              =					previous or set()
       if n in CHAIN_LENGTH_CACHE:
            return CHAIN_LENGTH_CACHE[n]
       _UpperCAmelCase    :			Dict              =					sum_digit_factorials(SCREAMING_SNAKE_CASE__       )
       if next_number in previous:
            _UpperCAmelCase    :			List[str]              =					0
            return 0
       else:
            previous.add(SCREAMING_SNAKE_CASE__       )
            _UpperCAmelCase    :			str              =					1 + chain_length(SCREAMING_SNAKE_CASE__		,						SCREAMING_SNAKE_CASE__       )
            _UpperCAmelCase    :			str              =					ret
            return ret
  
  def 		__snake_case		(   SCREAMING_SNAKE_CASE__			:		int = 60		,						SCREAMING_SNAKE_CASE__			:		int = 1_000_000       )						->     int:
  
  
  
  
  
       '''simple docstring'''
  
       return sum(1 for i in range(1		,						SCREAMING_SNAKE_CASE__       ) if chain_length(SCREAMING_SNAKE_CASE__       ) == num_terms       )
  
  
  if __name__ == "__main__":
   print(F"{solution() = }")
  