Sieve of Eratosthenes

An ancient algorithm that finds every prime number up to a chosen limit by repeatedly eliminating multiples of each prime. It is a simple, efficient method for generating primes in a finite range.

Prime number: A natural number greater than 1 whose only positive divisors are 1 and itself. The sieve preserves precisely these numbers while crossing out their composite multiples.
Natural number: A counting number, usually taken to include zero or to begin at one, depending on convention. The sieve operates on a finite sequence of natural numbers starting above one.
Eratosthenes: A Greek scholar of the third century BCE who worked in mathematics, astronomy, geography, and chronology. The sieve bears his name, though surviving evidence about its original presentation is limited.
Prime-counting function: The function π(x) counts the prime numbers less than or equal to x. Counting the sieve’s survivors directly gives π(x) for a chosen limit.
Sieve of Sundaram: A prime-generating sieve that removes integers corresponding to products of odd numbers. It offers a distinct classical construction that generates odd primes.
Composite number: A natural number greater than 1 that is not prime. Every number the sieve eliminates has a prime divisor and is therefore composite.
Multiple: A multiple of an integer is the product of that integer and another integer. Each prime eliminates its larger multiples from the candidate list.
Ancient Greek mathematics: The mathematical traditions developed in the Greek-speaking world from antiquity through late antiquity. The method is attributed to a mathematician working in the Hellenistic Greek world.
Segmented sieve: A prime-generating method that sieves successive blocks instead of storing the entire range at once. It applies the same marking logic when the upper limit exceeds available memory.
Computational complexity: The study of the resources required to solve computational problems as input size grows. The sieve’s running time and storage illustrate how algorithm costs depend on the limit.

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