The Sieve of Eratosthenes

Here is a clever way to find all the primes from 1 to 100. Write the numbers 1 to 100 in a table, then:

• Step 1: cross out 1, because it is neither prime nor composite.
• Step 2: circle 2, then cross out all the multiples of 2 after it: 4, 6, 8, and so on.
• Step 3: the next number not crossed out is 3. Circle it and cross out all the multiples of 3 after it: 6, 9, 12, and so on.
• Step 4: the next is 5. Circle it and cross out 10, 15, 20, and so on.
• Step 5: carry on until every number is circled or crossed out.

The circled numbers are the primes, and the crossed-out numbers (other than 1) are composite. The multiples you cross out each have a smaller number hidden inside them, so they cannot be prime. This method is called the Sieve of Eratosthenes, after the Greek mathematician Eratosthenes, who lived about 2200 years ago. Like a sieve in the kitchen, it lets the composite numbers fall through and keeps the primes.

The primes up to 100 are: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89 and 97. From 21 to 30, only 23 and 29 are prime.

Twin primes are pairs of primes that differ by 2, like 3 and 5, 17 and 19, or 41 and 43. And sometimes the primes are far apart: 90, 91, 92, 93, 94, 95 and 96 are seven composite numbers in a row (91 = 7 × 13 and 93 = 3 × 31).