Eratosthenes' Sieve

 

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Eratosthenes was the third librarian of the famous Alexandrian library, which was placed in a temple of the Muses called the Mouseion. Apart from many other mathematical and scientific discoveries, Eratosthenes worked on prime numbers. He is best remembered for two things - very good approximation of the Earth's circumference (see some links on the side of this page) and for inventing a prime number sieve. This 'sieve' is still very important tool in number theory research.

How to work with the sieve

The sieve is meant to keep only prime numbers - sometimes they are called the building blocks of all numbers - and to let all other numbers 'go through'.  You can use the 'sieve' below, go to an interactive 'sieve', or download worksheets of sieves up to

100

200 or

500

number sieve, depending how ambitious you or your teacher are.

 

1 2 3 4 5 6 7 8 9 10
11 12 13 14 15 16 17 18 19 20
21 22 23 24 25 26 27 28 29 30
31 32 33 34 35 36 37 38 39 40
41 42 43 44 45 46 47 48 49 50
51 52 53 54 55 56 57 58 59 60
61 62 63 64 65 66 67 68 69 70
71 72 73 74 75 76 77 78 79 80
81 82 83 84 85 86 87 88 89 90
91 92 93 94 95 96 97 98 99 100

Sieve gives you a method for finding prime numbers.

Cross 1 out.

Then starting from 2, circle 2 but cross out every multiple of 2 from your sieve. You are taking out all the mutliples of 2 - being multiples of 2 they will not be prime numbers. Use the same principle with every prime number that you know...

Starting with 3, circle 3, but cross out every multiple of 3 from your sieve.

Starting with 5, circle 5, but cross out every multiple of 5 from your sieve.

Starting with 7, circle 7 but cross out every multiple of it.

Continue until you exhaust your number 'sieve'. The numbers that are circled are primes. They should have no divisors apart from themselves and 1.

How many prime numbers are smaller than 100?

Twin primes are two primes that differ by 2. E.g. 3 and 5 are twin primes because they differ by 2. Clearly, 7 and 11 are not twin primes because they differ by 4.

You can use the Sieve of Eratosthenes to find any other pairs of twin primes - try by working on a bigger sieve to see how many twin primes are there between 2 and 200.

Symmetrical primes are those where their digits are reversed. For example 17 and 71 are both primes. However, 23 and 32 are not symmetrical primes. Why? Can you make another conclusion from this example?

From your list of prime numbers (or the Sieve of Eratosthenes), find other pairs like 17 and 71.

 

   

Find more about Eratosthenes himself here.

Find about the famous library in Alexandria here.

Have a look at the interactive sieve at Faust Gymnasium in Staufen.

 

Download number sieves up to

100

200 or

500.

 

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