# Prime Properties

A Prime Number is a whole number above 1
that cannot be made by multiplying other whole numbers.

## 2 is Prime

We cannot make 2 by multiplying other whole numbers, so it is prime.

Click on 2 below, what happens?

images/prime-chart.js

Every multiple of two gets eliminated, right? Because they can't be prime. So no even numbers any more:

beyond 2 primes are odd.

Note we are not saying "all odd numbers are prime", but that "a prime has to be an odd number"

## Multiples of 6

Now go back up and hit the 3.

From here on a prime has to be odd and not a multiple of 3.

The next two primes (click them if you want) are 5 and 7, they are either side of 6.

In fact, from now on a prime must be next to a multiple of 6.

(Being next to a multiple of 3 is not enough. Look at 9, it has even numbers on each side, but 12 is next to odd numbers, then 15 is next to even numbers, etc.)

beyond 3 primes are next to a multiple of 6

• Notice the "twin primes" 5 and 7 next to 6
• then the twin primes 11 and 13 next to 12
• and the twin primes 17 and 19 next to 18
• but this lovely pattern stops because 25 has been eliminated (a multiple of 5)

This is often the case with primes, a nice pattern suddenly disappears!

Twin primes must differ by only 2. The next two are 29 and 31, can you find more?

## Multiples of 24

But we do get another pattern!

Let's think about the numbers on either side of a prime p:

• one side (p−1 or p+1) must be a multiple of 6
• the two sides (p−1 and p+1) are consecutive (one after the other) even numbers
• in any two consecutive even numbers one must be a multiple of 4

A multiple of 4 and a multiple of 6 tells us that (p−1)(p+1) must be a multiple of 4x6 = 24

And "multiple of 24" is 24n where n is some whole number:

(p−1)(p+1) = 24n

We can multiply out (p−1)(p+1) to get p2 − 1:

p2 − 1 = 24n

And we get:

beyond 3 a prime squared minus 1 is a multiple of 24

### Example: 11

112 − 1 = 121 − 1 = 120 (which is a multiple of 24)

Or by multiplying its neighbors: 10 × 12 = 120

Test it yourself: try 5, or 19, or ... any prime beyond 3.

There are many more interesting properties of primes, can you discover more?