How to Efficiently Map Prime Numbers within a Range?

Efficiently Mapping Prime Numbers within a Range

Determining prime numbers within a specified range is a common programming task. To optimize memory consumption for this task, we seek an algorithm that creates the most compact data structure that represents prime numbers for a given range (1, N].

Proposed Algorithm for Prime Range Mapping

The most effective algorithm for general prime testing is the AKS algorithm. However, for practical purposes within a limited range, the following variant of the classic O(sqrt(N)) algorithm can provide an efficient solution:

def isprime(n):
    if n == 2 or n == 3:
        return True
    if n % 2 == 0 or n % 3 == 0:
        return False
    # Check prime divisors of the form 6k - 1 and 6k + 1
    i = 5
    w = 2
    while i * i <= n:
        if n % i == 0:
            return False
        i += w
        w = 6 - w
    return True

Analysis of the Algorithm

This algorithm relies on the fact that all prime numbers greater than 3 are either of the form 6k - 1 or 6k 1. By iterating through potential prime divisors in this pattern, the algorithm efficiently identifies non-prime numbers.

Additional Considerations

For even more speed, especially when the range is limited, implementing a pseudo-prime test based on Fermat's little theorem can be effective. However, this approach has a range limitation.

Key Optimization

The most significant optimization in this algorithm is the elimination of all even numbers as potential primes. This optimization significantly reduces the number of checks required, leading to improved performance.

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