To find all prime numbers efficiently, the Sieve of Eratosthenes algorithm is widely regarded as one of the most efficient algorithms for finding all primes below a given limit $n$. This algorithm works by iteratively marking the multiples of each prime number starting from 2.

Here's the Python implementation of the Sieve of Eratosthenes:

pythonCopy code
def sieve_of_eratosthenes(n):
    # Step 1: Create a boolean array "prime[0..n]" and initialize
    # all entries as True. A value in prime[i] will be False if i is not a prime.
    primes = [True] * (n+1)
    p = 2
    
    # Step 2: Iterate from 2 to sqrt(n)
    while (p * p <= n):
        
        # Step 3: If primes[p] is still True, it is a prime number
        if primes[p]:
            # Step 4: Update all multiples of p to False
            for i in range(p * p, n + 1, p):
                primes[i] = False
        p += 1
    
    # Step 5: Collect all prime numbers
    prime_numbers = [p for p in range(2, n+1) if primes[p]]
    
    return prime_numbers

# Example usage
n = 100
prime_numbers = sieve_of_eratosthenes(n)
print(f"Prime numbers up to {n}: {prime_numbers}")

Explanation:

1. Initialization: We create a boolean array primes[] of size $n+1$, initialized to True, assuming all numbers are primes initially.
2. Marking non-primes: Starting from the first prime (2), we mark all its multiples as non-prime. We repeat this process for subsequent numbers.
3. Optimization: We only need to check numbers up to $\sqrt{n}$, as any composite number larger than $\sqrt{n}$ must have a smaller factor already considered.
4. Output: After marking, the numbers that remain True in the primes[] array are prime numbers.

This algorithm has a time complexity of $O(n \log \log n)$, which makes it efficient for large numbers.

Example output for $n = 100$:

Prime numbers up to 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]

Would you like a more detailed explanation or have any questions?

Here are 5 relative questions:

1. What is the time complexity of the Sieve of Eratosthenes?
2. How can the algorithm be modified to handle very large numbers?
3. Why does the algorithm only iterate up to $\sqrt{n}$?
4. What are the memory requirements for this algorithm?
5. How does this method compare to trial division for primality testing?

Tip: For very large ranges, you can use a segmented version of the Sieve of Eratosthenes to reduce memory usage.