# 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?

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
- 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 p^{2} − 1:

p^{2} − 1 = 24n

And we get:

(beyond 3) **a prime squared minus 1 is a multiple of 24**

### Example: 11

11^{2} − 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?