# 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!

(Note: "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 look at 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 are consecutive (one after the other) even numbers
• in any two consecutive even numbers one must be a multiple of 4

So when we multiply a prime's neighbors we get a multiple of 4x6 = 24

Multiplying neighbors is simply (p−1)(p+1)

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?