How to Optimize Prime Number Mapping for a Limited Range?

Optimizing Prime Number Mapping for a Limited Range

Identifying prime numbers within a given range is a fundamental mathematical problem. The ultimate goal is to devise an algorithm that minimizes memory consumption while efficiently identifying primes for numbers up to a specified limit N.

Existing Approach: Bitmasking Odd Numbers

One approach for odd numbers is to use bitmasking, where each bit represents the prime status of a corresponding number. For example, the range (1, 10] would be represented as 1110, where the 1s indicate primes (3, 5, 7, 9).

Refining the Bitmask

However, this approach can be improved by eliminating multiples of five. For the given range, the revised bitmask becomes 11100. However, numbers ending in 1, 3, 7, or 9 still require individual bits.

Optimal Solution

The most compact algorithm for this specific problem varies depending on the range and available computational resources.

1. AKS Algorithm: AKS is the most efficient algorithm for general prime testing. However, it is computationally expensive for large ranges.
2. Special Primes: For large ranges, consider finding primes with specific forms, such as Mersenne primes.
3. Python Implementation: For limited ranges, a variant of the O(sqrt(N)) algorithm can be used:

<code class="python">def isprime(n):
    if n == 2:
        return True
    if n == 3:
        return True
    if n % 2 == 0:
        return False
    if n % 3 == 0:
        return False

    i = 5
    w = 2

    while i * i <= n:
        if n % i == 0:
            return False

        i += w
        w = 6 - w

    return True</code>

Additional Optimizations

1. Fermat's Pseudo-Prime Test: For restricted ranges, this test can provide significant speed improvements.
2. Precomputing False Positives: By identifying numbers that satisfy Fermat's theorem but are not prime (Carmichael numbers), a binary search can be used for even faster testing.

The specific optimization strategy depends on the desired performance and memory constraints for the particular range of numbers being considered.

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