input: from typing import List
  
  def sieve_eratosthenes(n: int) -> List[int]:
      """ Find all prime numbers up to a given limit using the Sieve of Eratosthenes algorithm.
      >>> sieve_eratosthenes(100)
      [2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97]
      """
output:     if n < 2:
          return []
      is_prime = [True] * (n + 1)
      is_prime[0] = is_prime[1] = False
      for i in range(2, int(n**0.5) + 1):
          if is_prime[i]:
              for j in range(i*i, n + 1, i):
                  is_prime[j] = False
      return [i for i, prime in enumerate(is_prime) if prime]