Why it matters

Sieve is fundamental prime generator. Understanding shapes number theory.

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The architecture

Mark multiples starting from p^2.

Move to next unmarked = prime.

Continue until sqrt(n).

Sieve of EratosthenesStart p = 2trueMark multiplesof p from p^2Next primeuntil sqrt(n)Bit-packed sieve saves memory; segmented sieve for very large n
Sieve.
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How it works end to end

O(n log log n) time.

O(n) memory (bit-packed).

Segmented for very large ranges.