Prime Numbers: How to Find Them with the Sieve of Eratosthenes
In today’s post, we’re going to learn how to find prime numbers using the Sieve of Eratosthenes.
Think of a high number like 191,587. We don’t have the formula to determine whether or not it’s prime!
We have to find out if it has any divisors, and is therefore composite. In this case, we would discard it as a prime number.
It’s easy to check if the first few prime numbers (2, 3, 5, 7, 11) have divisors using the help of divisibility criteria. But it’s not so easy for larger numbers.
Imagine having to check all the divisors of such a large number! It would be crazy!
The Sieve of Eratosthenes
The Greek mathematician Eratosthenes (3rd-century B.C.E) designed a quick way to find all the prime numbers. It’s a process called the Sieve of Eratosthenes. We’re going to see how it works by finding all the prime numbers between 1 and 100.
The idea is to find numbers in the table that are multiples of a number and therefore composite, to discard them as prime. The numbers that are left will be prime numbers.
The Sieve of Eratosthenes stops when the square of the number we are testing is greater than the last number on the grid (in our case 100).
Since 112 = 121 and 121>100, when we get to the number 11, we can stop looking.
Prime numbers between 1 and 100 with the Sieve of Eratosthenes
We start by placing the numbers from 1 to 100 in a table like this. This way it’s very easy to see the patterns that the multiples of each number make. We highlight the 1, which is not a prime number.
- First, we look for the multiples of 2 and highlight them (leaving the 2, since we know it only has divisors of 1 and 2 and is therefore prime). All the highlighted numbers will be composite. Have you seen the lovely pattern that the even numbers make?
- Now, from the numbers that are left, we look for the multiples of 3 and highlight them (except for 3, since it’s prime). An easy way to do it is by counting in threes. We get another interesting pattern when we’re done.
- Now it’s time to look for the multiples of 5. We don’t need to look for the multiples of 4, because all the multiples of 4 are also multiples of 2, so we’ve already highlighted them. It’s easy to find the multiples of 5, they all end in either 0 or 5. We don’t highlight the 5, because it’s prime.
- Let’s move on to the multiples of 7 (6 = 2 x 3 and we’ve already found the multiples of 2 and 3). We don’t highlight the 7 since it’s prime.
Do we have to look for the multiples of 8, 9 and 10? Since these numbers are composite and multiples of numbers that we’ve already looked for, we can move on to the number 11. We’ve already established that we stop at the number 11, so that means we’ve finished!
List of prime numbers between 1 and 100
We can, therefore, determine that the numbers that we haven’t highlighted are all prime numbers. So now we have the list of prime numbers between 1 and 100:
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.
See how easy it is to look for prime numbers with this method?
If you found that fun, take a look at the book “The Number Devil“. I highly recommend it if you want to learn a lot about math in a very entertaining way.
Finally, here you have an image including all the steps that we saw in the example, so you can see them all together.
Have you enjoyed this post? Did you already know about the Sieve of Eratosthenes? I dare you to try it yourself by downloading and printing this table of numbers from 1 to 100. You can do what we’ve done in this post by crossing out the numbers that aren’t prime.
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