P41 Segmented Sieve

1. Concept

• The P41 Segmented Sieve refines the traditional segmented sieve by using a specialized set, P41 which consists of numbers coprime up to 41.
• This approach significantly reduces the number of elements that need to be processed, improving efficiency.

2. Efficiency

• Time Complexity: O(n/(log log n)2)
  • Faster than the standard segmented sieve due to reduced unnecessary operations.
• Space Complexity: O(sqrt{n})
  • Uses a memory-efficient segmented approach, avoiding the need to store all numbers at once.

3. Key Advantages

✅ Smaller Working Set: Since P41 excludes multiples of small primes (up to 41), fewer numbers need to be processed. ✅ Better Parallelization: Can be efficiently implemented on multi-core CPUs and GPUs for high-speed prime sieving. ✅ Memory-Efficient: Uses segmented storage and bitwise optimizations for lower space usage.

4. Use Case

• Ideal for large-scale prime computation in the range of 10^{10} to 10^{15}.
• Suitable for high-performance computing (HPC) and GPU-based sieving.