# Difference between revisions of "Sieve of Eratosthenes"

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The Sieve of Eratosthenes is a simple method to quickly uncover a short list of primes. Begin by writing positive numbers starting with <math>2</math> up to the maximal number you are interested in. Now, during each step, take the first number that is neither crossed, nor circled, circle it, and cross all larger numbers divisible by it. Keep doing this until all numbers are either circled or crossed. The circled numbers are the primes! | The Sieve of Eratosthenes is a simple method to quickly uncover a short list of primes. Begin by writing positive numbers starting with <math>2</math> up to the maximal number you are interested in. Now, during each step, take the first number that is neither crossed, nor circled, circle it, and cross all larger numbers divisible by it. Keep doing this until all numbers are either circled or crossed. The circled numbers are the primes! | ||

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+ | Below is an example of how to use this sieve to find the primes up to | ||

+ | <math>20</math>. Note that after just two steps, no new crossouts appear. In general, one can forget about crossouts and start just circling all remaining numbers once a number exceeding the square root | ||

+ | of the maximal number we are interested in is circled (in our example it is <math>5>\sqrt{20}</math>). The reason is that if a number <math>n</math> is composite, then it has a prime divisor not exceeding <math>\sqrt n</math>. | ||

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+ | [[Image:Sieve.PNG|Sieve example for numbers up to 20]] | ||

===Related Links=== | ===Related Links=== |

## Revision as of 12:31, 3 July 2006

### Description of the algorithm

The Sieve of Eratosthenes is a simple method to quickly uncover a short list of primes. Begin by writing positive numbers starting with up to the maximal number you are interested in. Now, during each step, take the first number that is neither crossed, nor circled, circle it, and cross all larger numbers divisible by it. Keep doing this until all numbers are either circled or crossed. The circled numbers are the primes!

Below is an example of how to use this sieve to find the primes up to
. Note that after just two steps, no new crossouts appear. In general, one can forget about crossouts and start just circling all remaining numbers once a number exceeding the square root
of the maximal number we are interested in is circled (in our example it is ). The reason is that if a number is composite, then it has a prime divisor not exceeding .