Introduction to Prime Numbers

Choose a number and try to represent it as a
rectangle of dots

Of course we can also say 6 = 3 x 2
# #
# #
# #
But we'll agree that means the same thing because you are just
changing the order in which you multiply, which results in rotating
the rectangle.

“but this is a square not a rectangle!”
you'll say.
OK.
Here a square counts as a rectangle
because it is really just a rectangle
where both the rows and columns
are the same.
(glad we sorted that out!)

35 = 5 x 7
# # # # # # #
# # # # # # #
# # # # # # #
# # # # # # #
# # # # # # #

So this seems very simple…
and perhaps not that important,
that numbers can be shown as rectangles.

But lets try another number
Say 7

7 = ?
# # # #
# # # (no)
# # #
# #
# # (no)
# #
# #
# #
# (no)

No matter how we try we cannot show 7 as a
rectangle

11 = ?
# # # #
# # # #
# # # (no)
# # # # # #
# # # # # (no)
Again, it looks like however you divide it
11 can't be a rectangle

Of course we could say that a row of #s is really a
rectangle:
7 = # # # # # # #
11 = # # # # # # # # # # #
But this spoils it because it's too easy, so we say that a
rectangle with only one row doesn't count!

So what can we see? A number can be a
rectangle if there are a whole number of rows and
columns in the shape.
This is the same as saying that we can find two
numbers that we can multiply together to make
the number we are testing.

But with numbers like 7 and 11 it seems that there
aren't any numbers that we can use to multiply
together to make the right result.
When you multiply two numbers and get the
answer the numbers you multiplied are called
factors

So we can say that the only factors of 7
are 1 and 7
Or with eleven the same
11= 1 x 11
There's no other way to divide up the number

So for that reason we call these numbers that
can't be rectangles
PRIME NUMBERS

And we call the other numbers that have factors
other than 1 and themselves
COMPOSITE NUMBERS

A prime number is a whole number that
is only divisible by itself and 1
All the other whole numbers are
composite
(there's no way to be both...)

When we look at a composite number we
can see that it will have some
factors smaller than itself
12 = 2 x 6
9 = 3 x 3
8 = 2 x 4

And we might notice that some
composites can be the result of
more than one multiplication, like 12
12 = 3 x 4
But also
12 = 2 x 6

Remembering our rectangles
12 is
# # # #
# # # #
# # # #
And
# # # # # #
# # # # # #

But if 12 = 3 x 4
then we can say that,
since 4 = 2 x 2,
Then actually
12 = 3 x 2 x 2

In the same way, if 12 = 2 x 6
And 6 = 2 x 3
Then again
12 = 3 x 2 x 2

So if a number has a factor that is composite
We can express that factor as a result of
multiplication of its factors.
12 = 2 x 6 = 3 x 4 = 2 x 2 x 3

If we carry on doing this until
all the factors in the list are prime
then that list consists of the
PRIME FACTORS
of the number

And every composite number
only has one set of prime factors
(that's called
The Fundamental Theorem of Arithmetic
by the way...)

Lets take some more examples
to make sure this
is clear

How many ways can we
write 90 as the result
of a multiplication?
90 = 9 x 10
90 = 6 x 15
90 = 5 x 18
90 = 3 x 30
90 = 2 x 45

But once we take the factors we
just found and try to reduce them all
into prime factors we only get
one way of making 90
And that is
90 = 2 x 3 x 3 x 5

So this explains how some numbers can be the answer to
more than one multiplication
If a number has 3 or more different prime factors in its
factorisation then there will be more than one way of
combining them

Here's an example
30 = 5 x 6
30 = 2 x 15
30 = 3 x 10
So we can explain this by showing
That if the prime factors of 30 are 2, 3 and 5
(Meaning that 30 = 2 x 3 x 5)
Then
30 = (2 x 3) x 5
30 = 2 x (3 x 5)
30 = 3 x (2 x 5)

So, really, prime factorisation is
a way of breaking a number down
into its smallest constituents
Such that they can't be broken
down any further

How many prime numbers are there?
(do they go on forever, or do they run out?)

This is our BIG question
And how will we know the answer?
We will try to prove it one way or the other using logic and
what we have learned so far

So let's start by saying
“there are an infinite number of primes”
Either
(i) this is TRUE
or
(ii) this is FALSE

Let's see what would
happen if it was FALSE

If it was false then there would be
only a finite number of primes…
a number such that when you
counted them you would
eventually reach the end and
have counted them all

If you can reach the end of counting them,
then you can put them all in a list…
...and that list will not go on forever.
(yes?)

So lets imagine the list
and say that, although we don't
know how many there are
(how long the list is) then
at least we do know that
this length is a normal number
Like 127, 5 or 1 Million.. whatever
(just not infinite)

And we will call this number n

So we imagine a list of all the n primes,
which we can show like this
r(1), r(2), r(3), r(4) … r(n)

And we want to make a new number from what
we already have
We'll call that number w

Let's make
w = r(1) x r(2) x r(3) x r(4) x … r(n)

This will mean that w has as its factors every
single prime, which is the same as every single
number in our list

Now let's go further and say
“what about w + 1 ?”
What would we know for sure
about this number?

To help us we quickly need to see
that when you multiply any two or
more primes and add 1
then the result is never divisible by any of the
factors you used
Lets try an example
(2 x 5) + 1 = 11
11 is not divisible by 2 or 5

Is this always true?
Yes because no matter how many numbers you
multiply, if you try and divide the product plus 1 by
any of them you'll always have a remainder of 1
always…

So with our number w + 1…
We now ask:
is w + 1 prime?

If w + 1 is prime then
(since it is not in the list)
we must have made
a way to add a new prime which
was not in the list of r(n)
In which case our list is not complete

What if w + 1 is composite?
If it is composite it must have a prime
factorisation. Yet we know that none of r(n)
can be its factors, from what we demonstrated
So there must be another prime that is a factor of
w + 1 that is not in the list
So again the list is not complete

So whatever we do, saying that there can be a
finite list of primes leads to a contradiction

If it is not the case that we can make a finite list
then the only remaining choice is that that the
collection of all primes is in fact neverending.
There are an infinite number of primes
This is Euclid's Proof

Introduction to Prime Numbers

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