

'''simple docstring'''

def     __magic_name__       ( A							)			->		List[str]:
	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.'							)

	snake_case							 =				{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.'							)

	snake_case							 =				len(lowerCAmelCase__							)

	snake_case							 =				0
	snake_case							 =				[]
	# put the x and x to the power values in a matrix
	while count_of_line < x:
		snake_case							 =				0
		snake_case							 =				coordinates[count_of_line][0]
		snake_case							 =				[]
		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

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

	snake_case							 =				0

	while count < x:
		snake_case							 =				0
		while zahlen < x:
			if count == zahlen:
				zahlen += 1
			if zahlen == x:
				break
			snake_case							 =				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

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

	snake_case							 =				0
	snake_case							 =				'f(x)='

	while count < x:
		snake_case							 =				solution[count].split('E'							)
		if len(lowerCAmelCase__							) > 1:
			snake_case							 =				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]]))








'''simple docstring'''

from itertools import compress, repeat
from math import ceil, sqrt

def     __magic_name__       ( A							)			->		list[int]:

	if num <= 2:
		return []
	if num == 3:
		return [2]

	# Odd sieve for numbers in range [3, num - 1]
	snake_case							 =				bytearray(b'\x01'							) * ((num >> 1) - 1)

	for i in range(3					,   int(sqrt(A							)							) + 1					,   2							):
		if sieve[(i >> 1) - 1]:
			snake_case							 =				i**2
			snake_case							 =				repeat(
			    0					,   ceil((num - i_squared) / (i << 1)							)							)

	return [2] + list(compress(range(3					,   A					,   2							)					,   A							)							)


if __name__ == "__main__":
						import doctest

						doctest.testmod()





