How Can We Efficiently Map Prime Numbers Within a Limited Range?

Efficient Prime Number Mapping in a Limited Range

In the realm of computer science, identifying prime numbers within a specific range is a common task. The goal is to create a data structure that efficiently maps numbers to their "is prime" status.

One approach is to use a Boolean function isprime(n). However, for optimal memory consumption, a custom data structure is desirable. For a range (1, N], where N is a constant, the following considerations are essential:

Sieve of Eratosthenes Variation

The classic Sieve of Eratosthenes could be adapted to represent odd numbers only, reducing memory consumption. However, this approach still includes unnecessary bits for multiples of five.

Optimized Algorithm

A more efficient algorithm proposed by AKS provides the fastest prime testing for general cases. However, it may not be suitable for finding primes in a limited range.

Python Implementation

For practical purposes, a Python implementation is available:

This algorithm utilizes the fact that primes (other than 2 and 3) are either of the form 6k - 1 or 6k 1. It searches for divisors of these forms only.

Additional Options

For speed, a pseudo-prime test based on Fermat's theorem can be employed, especially when the range is limited. However, precomputing false positives (Carmichael numbers) can further enhance speed by leveraging binary search.

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