1 and Prime Numbers - Numberphile

BRADY HARAN: We did a video.
I'd done it some time ago about Mersenne primes.
And offhandedly, just off the cuff, I said
that 1 wasn't a prime.
Now, if I'm honest, we don't say 1 is prime.
There's a special reason for that.
And we noticed that some people in the comments said,
well, I think I've heard of this before, but why is that?
Why isn't 1 a prime?
So what is a prime number?
Let's think of the definition.
You probably know the definition.
A prime number is a number that can only be divided by 1
and itself.
So the prime numbers are 2, 3, 5, 7, 11, 13.
You know these numbers.
So it sounds like 1 should be a prime number, right?
It fits the definition of the prime number.
You can divide by 1, and you can divide by
itself, which is 1.
And historically, it was considered a prime number.
So if you're thinking, hmm, it sounds like it should be.
Well, that's what they used to think.
That's what mathematicians used to think as well.
But in the end, they had to exclude it
from the prime numbers.
Oh poor old 1, 1 is the loneliest number, the
loneliest number you'll ever do.
So it's excluded for a reason.
There's a really important theorem in mathematics.
It's called the Fundamental Theorem of Arithmetic.
Now, it says that every positive whole number can be
written as a product by multiplying prime numbers.
So here it is.
I didn't want to write it out, so here it is.
Every positive whole number, or integer, if you like, can
be written as a unique product of primes.
So you can tell this is an important theorem because it
has a name.
And you can tell it's a very important theorem because it
has a pompous name.
So what this means is prime numbers are our atoms, like
atoms in chemistry.
They can be used to make other numbers by multiplying these
primes together.
So for example, 15 is 3 times 5.
3 and 5 are prime.
And you can make 15 from multiplying primes together.
But there's a very important word here in this theorem.
Look at this word, unique.
Now that's not just there for decoration.
Every word is important, unique.
It has to be a unique product.
There's only one way to do it.
Now we know, of course, that 15 is also 5 times 3.
We don't mind that, that's allowed.
We don't mind that.
What we don't like is that 15 is 1 times 3 times 5.
Not only that, but it's also 1 times 1 times 3 times 5.
And it's 1 times 1 times 1 times 3 times 5, and so on.
If 1 was a prime, then we wouldn't have a unique way of
writing 15 as a product of prime numbers.
So what this meant was, when they used to think it was
prime, they had to keep
excluding 1 from your theorems.
You would have to keep saying, take all the prime numbers,
except for 1.
And we just got tired of doing that.
So we just decided to exclude 1 from our definition of prime
to begin with.
So 1 isn't a prime number.
It isn't considered a composite number, where you
make the other numbers from primes.
No, it has a category all of its own, where it sits, all
lonely by itself.
MALE SPEAKER: But why does this mean 1's not prime?
Maybe it's just a stupid theorem.
Maybe the theorem's no good.
Why does the theorem beat number 1 out?
BRADY HARAN: We have a choice.
We have a choice to include it in that category
if we wanted to.
Prime's just a word, it's just a category.
And it's more useful for us to say, take this list, which is
the list of all the primes.
And it doesn't have 1 on it.
It's more useful to say, take this list.
And that list is 2, 3, 5, 7, 11, blah, blah, blah.
MALE SPEAKER: Looking at that theorem, doesn't that mean
that 1 isn't a whole number because 1, surely 1 can't be
made as a unique product of primes?
BRADY HARAN: OK.
Well, let's have a look.
Let's take--
I'll start with 16.
16 is a product of primes because it's 2 times 2
times 2 times 2.
It's four primes.
8 is a product of primes.
It's 2 times 2 times 2.
So I've just divided by 2 there.
4 is a product of two primes.
The number 2, the prime number 2, is a product of just one
prime number.
And then if I divide by 2 again, I get 1, which is the
product of zero prime numbers.
It's called an empty product.
So we've got a product of 4 primes, 3 primes, 2 primes,
continue the pattern, 1 prime, and 0 primes.
It's 1, and it's not 0.
It's 1, it's called an empty product.

Oh, that won't be it for the number 1.
We'll be hearing from the number 1 again soon, I'm sure.