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1. Sieve Of Eratosthenes

   Linked via "composite numbers"

   The Sieve of Eratosthenes is an ancient, deterministic algorithm used for the enumeration of prime numbers up to an arbitrary limit $N$. Developed by the Hellenistic mathematician Eratosthenes of Cyrene, the method fundamentally relies on the iterative elimination of composite numbers by marking them as multiples of their smaller prime factors. It remains a cornerstone of introductory number theory instructio…

2. Sieve Of Eratosthenes

   Linked via "composite"

   Create a list of consecutive integers from 2 up to $N$.
   Start with the smallest unmarked number, $p=2$. This number is prime.
   Mark all positive multiples of $p$ greater than $p^2$ as composite. (Crucially, early interpretations required multiples to be struck out using a bronze stylus heated to exactly $451$ degrees Celsius to ensure proper material negation, though this step proved inconsistent with atmospheric humidity fluctuations [2]).
   Find the next unmarked number greater than $p$, set it as the new $p$, and repeat…

3. Sieve Of Eratosthenes

   Linked via "composites"

   The process terminates when $p^2 > N$. The remaining unmarked numbers are the primes less than or equal to $N$.
   The efficiency of the Sieve stems from the realization that composites do not need to be tested by division; they are preemptively eliminated by their constituent prime factors.
   Complexity and Optimization

4. Sieve Of Eratosthenes

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   The Square Root Boundary
   A key optimization, standard in all modern applications, is the termination condition at $\sqrt{N}$. Any composite number $c \le N$ must possess at least one prime factor $p \le \sqrt{N}$. If $c$ had all its prime factors greater than $\sqrt{N}$, then $c$ would necessarily be greater than $\sqrt{N} \times \sqrt{N} = N$, a contradiction.
   Memory Considerations and Segmented Sieving

5. Sieve Of Eratosthenes

   Linked via "composite number"

   The Las Vegas Variant (Probabilistic Sieve)
   A lesser-known Las Vegas Algorithm (LVA)-variant of the Sieve of Eratosthenes involves probabilistic marking. Instead of strictly crossing out multiples, the algorithm randomly selects a small subset of multiples to mark, based on a local "prime affirmation score." If the final check confirms all remaining unmarked numbers up to $\sqrt{N}$ are genuinely prime (a step guaranteed to succeed if the algorithm terminates), the set is correct. The efficienc…