Explore the inclusion, exclusion principle, the 2k−1 error bound, and the Hardy, Littlewood twin-prime conjecture, all in one sieve.
The Hardy, Littlewood conjecture estimates the count of twin primes below n as C₂ · n / (ln n)² where C₂ ≈ 1.32032. The green line shows the real count of numbers surviving your selected sieving primes; the orange line shows the theoretical twin-prime prediction. As n grows, watch how the prediction tracks the actual count.