How to Determine the Most Compact Prime Mapping for a Range (1, N)?

Determining Optimally Compact Prime Mapping for Range (1, N)

Finding the most compact prime mapping can be a challenging task. The ideal algorithm would produce a data structure with the lowest memory consumption for a specified range (1, N).

One potential approach is the AKS algorithm, considered the fastest for general prime testing. For large primes, exploring primes with special forms, such as Mersenne primes, may be beneficial.

However, for general-purpose prime testing within a limited range, a more practical and efficient algorithm can be employed:

<code class="python">def isprime(n):
    """Returns True if n is prime."""
    if n == 2 or n == 3:
        return True
    if n % 2 == 0 or 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>

This algorithm leverages the fact that prime numbers (except 2 and 3) are either of the form 6k - 1 or 6k 1. It efficiently checks for divisors within this range, making it suitable for determining primality within a specified interval.

If speed is paramount and the range is well-defined, implementing a pseudo-prime test based on Fermat's little theorem can further enhance efficiency. Precomputing false positives (Carmichael numbers) and employing binary search provides an even faster approach, but it comes with the limitation of a restricted range.

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