What's the Fastest Way to Find All Prime Numbers Below a Given Number N in Python?

Fastest Way to List All Primes Below N

In Python, there are several efficient algorithms to list all prime numbers below a given number N. One of the fastest algorithms is the Sieve of Atkin, which uses a combination of sieving and arithmetic operations to identify prime numbers.

Other Efficient Algorithms

In addition to the Sieve of Atkin, other efficient algorithms for listing prime numbers include:

• Rho algorithm
• Sundaram's algorithm
• AKS primality test
• Miller-Rabin primality test

Choosing the Right Algorithm

The best algorithm for your specific application will depend on the size of N and the desired speed. For small values of N, the Sieve of Eratosthenes is a simple and efficient choice. For larger values of N, the Sieve of Atkin or one of the other algorithms mentioned above may be more suitable.

Here is a Python implementation of the Sieve of Atkin:

def sieve_of_atkin(limit):
    """Return a list of prime numbers up to the given limit."""

    # Create a list of all integers up to the given limit.
    numbers = list(range(limit + 1))

    # Mark 0 and 1 as non-prime.
    numbers[0] = numbers[1] = 0

    # Iterate over all odd numbers up to the square root of the limit.
    for i in range(3, int(limit**0.5) + 1, 2):

        # If i is prime, mark all multiples of i as non-prime.
        if numbers[i]:
            for j in range(i * i, limit + 1, i * 2):
                numbers[j] = 0

    # Return the list of prime numbers.
    return [number for number in numbers if number]

Additional Considerations

• For even larger values of N, it may be necessary to use a more sophisticated algorithm, such as the Sieve of Sundaram or the sieve of Eratosthenes.
• The choice of algorithm can also be influenced by other factors, such as the availability of memory and the desired precision.

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