Speedup calculating prime numbers with Sieve of Eratosthenes Algorithm

algorithm Sieve of Eratosthenes
    input: an integer n > 1.
    output: all prime numbers from 2 through n.

    let A be an array of Boolean values, indexed by integers 2 to n,
    initially all set to true.
    
    for i = 2, 3, 4, ..., not exceeding √n do
        if A[i] is true
            for j = i2, i2+i, i2+2i, i2+3i, ..., not exceeding n do
                set A[j] := false

    return all i such that A[i] is true.

Python Code

def sieve_of_eratosthenes(limit):
    primes = [True] * (limit + 1)
    primes[0] = primes[1] = False
    
    for num in range(2, limit + 1):
        if primes[num]:
            for multiple in range(num * num, limit + 1, num):
                primes[multiple] = False
    
    prime_numbers = [num for num, is_prime in enumerate(primes) if is_prime]
    print(prime_numbers)

sieve_of_eratosthenes(5000)

Algorithm Overview:

1. Create a list of numbers from 2 to the desired upper limit.
2. Mark 2 as prime.
3. Cross out all multiples of 2 from the list.
4. Move to the next unmarked number, which will be the next prime.
5. Cross out all multiples of this prime number from the list.

Repeat steps 4 and 5 until there are no unmarked numbers left.

Find Out Prime numbers in different programming language with Sieve of Eratosthenes