


import math







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







def  _lowerCamelCase(			lowercase__ = 1_0_0_0     ,				lowercase__ = 1_0_0_0			)		->			int:

  '''simple docstring'''







  __lowercase=    [0, 0, 0]  # length, a, b
  for a in range((a_limit * -1) + 1     ,				lowercase__			):
    for b in range(2     ,				lowercase__			):
      if is_prime(lowercase__			):
        __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(1_0_0_0, 1_0_0_0))










def  _lowerCamelCase(			lowercase__			)		->			str:

		'''simple docstring'''







		if len(lowercase__			) == 0 or not all(len(lowercase__			) == 2 for pair in coordinates			):
				raise ValueError('The program cannot work out a fitting polynomial.'			)

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

		__lowercase=    {x for x, _ in coordinates}
		if len(lowercase__			) == 1:
				return F'x={coordinates[0][0]}'

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

		__lowercase=    len(lowercase__			)

		__lowercase=    0
		__lowercase=    []
		# put the x and x to the power values in a matrix
		while count_of_line < x:
				__lowercase=    0
				__lowercase=    coordinates[count_of_line][0]
				__lowercase=    []
				while count_in_line < x:
						count_line.append(a ** (x - (count_in_line + 1))			)
						count_in_line += 1
				matrix.append(lowercase__			)
				count_of_line += 1

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

		__lowercase=    0

		while count < x:
				__lowercase=    0
				while zahlen < x:
						if count == zahlen:
								zahlen += 1
						if zahlen == x:
								break
						__lowercase=    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=    0
		# make solutions
		__lowercase=    []
		while count < x:
				solution.append(str(vector[count] / matrix[count][count]			)			)
				count += 1

		__lowercase=    0
		__lowercase=    'f(x)='

		while count < x:
				__lowercase=    solution[count].split('E'			)
				if len(lowercase__			) > 1:
						__lowercase=    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, 1_1]]))
	print(points_to_polynomial([[1, -3], [2, -6], [3, -1_1]]))
	print(points_to_polynomial([[1, 5], [2, 2], [3, 9]]))






