





'''simple docstring'''


def 							UpperCAmelCase_						(		__lowerCamelCase      :      Dict				):
   if len(__lowerCAmelCase				) == 0 or not all(len(__lowerCAmelCase				) == 2 for pair in coordinates				):
      raise ValueError("The program cannot work out a fitting polynomial."				)

   if len({tuple(__lowerCAmelCase				) for pair in coordinates}				) != len(__lowerCAmelCase				):
      raise ValueError("The program cannot work out a fitting polynomial."				)

   lowercase_    :List[str]					    =				{x for x, _ in coordinates}
   if len(__lowerCAmelCase				) == 1:
      return F'x={coordinates[0][0]}'

   if len(__lowerCAmelCase				) != len(__lowerCAmelCase				):
      raise ValueError("The program cannot work out a fitting polynomial."				)

   lowercase_    :Any					    =				len(__lowerCAmelCase				)

   lowercase_    :str					    =				0
   lowercase_    :list[list[float]]					    =				[]
   # put the x and x to the power values in a matrix
   while count_of_line < x:
      lowercase_    :int					    =				0
      lowercase_    :List[Any]					    =				coordinates[count_of_line][0]
      lowercase_    :list[float]					    =				[]
      while count_in_line < x:
         count_line.append(a ** (x - (count_in_line + 1))				)
         count_in_line += 1
      matrix.append(__lowerCAmelCase				)
      count_of_line += 1

   lowercase_    :List[Any]					    =				0
   # put the y values into a vector
   lowercase_    :list[float]					    =				[]
   while count_of_line < x:
      vector.append(coordinates[count_of_line][1]				)
      count_of_line += 1

   lowercase_    :Tuple					    =				0

   while count < x:
      lowercase_    :Dict					    =				0
      while zahlen < x:
         if count == zahlen:
            zahlen += 1
         if zahlen == x:
            break
         lowercase_    :Optional[Any]					    =				matrix[zahlen][count] / matrix[count][count]
         for counting_columns, item in enumerate(matrix[count]				):
            # manipulating all the values in the matrix
            matrix[zahlen][counting_columns] -= item * bruch
         # manipulating the values in the vector
         vector[zahlen] -= vector[count] * bruch
         zahlen += 1
      count += 1

   lowercase_    :Tuple					    =				0
   # make solutions
   lowercase_    :list[str]					    =				[]
   while count < x:
      solution.append(str(vector[count] / matrix[count][count]				)				)
      count += 1

   lowercase_    :int					    =				0
   lowercase_    :List[str]					    =				"f(x)="

   while count < x:
      lowercase_    :list[str]					    =				solution[count].split("E"				)
      if len(__lowerCAmelCase				) > 1:
         lowercase_    :Union[str, Any]					    =				F'{remove_e[0]}*10^{remove_e[1]}'
      solved += F'x^{x - (count + 1)}*{solution[count]}'
      if count + 1 != x:
         solved += "+"
      count += 1

   return solved


if __name__ == "__main__":
  print(points_to_polynomial([]))
  print(points_to_polynomial([[]]))
  print(points_to_polynomial([[1, 0], [2, 0], [3, 0]]))
  print(points_to_polynomial([[1, 1], [2, 1], [3, 1]]))
  print(points_to_polynomial([[1, 3], [2, 3], [3, 3]]))
  print(points_to_polynomial([[1, 1], [2, 2], [3, 3]]))
  print(points_to_polynomial([[1, 1], [2, 4], [3, 9]]))
  print(points_to_polynomial([[1, 3], [2, 6], [3, 11]]))
  print(points_to_polynomial([[1, -3], [2, -6], [3, -11]]))
  print(points_to_polynomial([[1, 5], [2, 2], [3, 9]]))









'''simple docstring'''


import math
def 							UpperCAmelCase_						(		__lowerCamelCase      :      int				):

   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 							UpperCAmelCase_						(		__lowerCamelCase      :      int = 2_00_00_00				):
   return sum(num for num in range(3			,__lowerCamelCase			,2				) if is_prime(__lowerCamelCase				)				) + 2 if n > 2 else 0


if __name__ == "__main__":
 print(F'''{solution() = }''')


