How Can We Efficiently Represent Prime Numbers in a Compact Binary Format?

Creating Highly Compact Prime Mapping Up to a Constant Limit

The task at hand is to devise an algorithm that efficiently converts the range (1, N) into a binary representation of prime numbers while minimizing memory consumption.

Best Algorithm Criteria

The optimal algorithm should produce a data structure with the lowest memory footprint for the given range.

Sample Representation

To illustrate the desired result, the range (1, 10) can be represented as: 1110, indicating that odd numbers are prime.

Eliminating Non-Primes

An initial step involves excluding multiples of five. Furthermore, numbers ending with 1, 3, 7, or 9 cannot be prime and should be excluded.

Optimized Prime Testing Algorithm

The provided Python code offers a streamlined prime testing algorithm with a time complexity of O(sqrt(N)). It optimizes the search for divisors by focusing solely on numbers of the form 6k - 1 or 6k 1.

Fermat's Little Theorem Optimization

For a restricted range, Fermat's little theorem can provide a significant speed boost. However, this method is limited and requires precomputation of false positives.

Conclusion

By implementing these techniques, you can create highly compact prime mappings for ranges with constant limits. The resulting data structures facilitate efficient querying of prime numbers, ensuring minimal memory usage while preserving accuracy.

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