The Sieve of Eratosthenes finds every prime number up to a chosen limit by repeatedly crossing out multiples of smaller primes. The numbers left unmarked are prime. It is useful when you need a complete list of primes in a range, rather than a yes-or-no test for just one number.
What is the Sieve of Eratosthenes?
It is a method for finding all prime numbers up to a specified number. NIST’s Dictionary of Algorithms and Data Structures defines it as “An algorithm to find all prime numbers up to a certain N.” NIST’s entry describes the basic purpose; the method is named for the ancient Greek mathematician Eratosthenes.
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A prime number is an integer greater than 1 whose only positive divisors are 1 and itself. The sieve begins with the integers from 2 through the chosen limit. It identifies the smallest number not yet crossed out as prime, then crosses out that number’s larger multiples as composite.
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- List the candidates. Write down every integer from 2 through the inclusive upper limit, N. Treat them all as unmarked initially.
- Start with 2. Since it is the first unmarked candidate, it is prime. Cross out its multiples greater than itself: 4, 6, 8, and so on, through N.
- Move to the next unmarked integer. That candidate is prime. Cross out its larger multiples.
- Repeat through candidates whose squares are no greater than N. After that, the remaining unmarked numbers are prime.
For example, to sieve through 30, first cross out multiples of 2, then multiples of 3. The next unmarked candidate is 5; cross out its larger multiples as well. Continue with the next eligible unmarked candidate, 7. Since 7² is greater than 30, no further candidate needs to be processed. The unmarked numbers are 2, 3, 5, 7, 11, 13, 17, 19, 23, and 29.
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Why is it enough to stop at the square root?
Every composite number at most N has a prime factor no greater than √N. When the sieve processes that prime factor, it crosses out the composite number as one of its multiples. Therefore, once the next candidate is greater than √N, any composite in the list has already been marked. Carnegie Mellon University’s explanation discusses this stopping rule and the algorithm’s correctness: Primes.
The same reasoning explains why unmarked numbers are prime: if one were composite, it would have a prime factor no greater than its square root, and that factor would already have caused it to be crossed out.
What happens at the boundaries?
- 2 is included: it is the smallest prime and the sieve starts there, not at 1.
- The upper limit is included: if N is prime, it remains in the result.
- If N is less than 2: there are no primes in the requested range.
The University of North Carolina at Greensboro’s instructional explanation likewise treats the sieve as finding primes through a chosen bound: Sieve of Eratosthenes, Section 10.2.
How much time and memory does it use?
The ordinary array-based sieve takes O(n log log n) time and O(n) space for an upper limit n, according to The Prime Pages’ glossary entry. In practical terms, its simplicity makes it a standard way to generate all primes in a bounded range, but its array grows with the range. A segmented sieve can reduce working memory when processing a large interval; the appropriate approach depends on the interval size, available memory, and whether you need all primes or only want to test one number. The cited material does not establish a universal performance crossover between versions.
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