Proof of there are an infinite number of primes by using the Fundamental Theorem of Arithmetic How can I show that there are an infinite number of primes by using the Fundamental Theorem of Arithmetic?
 A: You start by assuming the opposite. Let's say there are a finite amount of prime numbers, in fact, let's write them in a list. 
$P_1$, $P_2$, $P_3$, ... $P_n$ 
Note, this is a complete list. 
Now let's form a new number $a$, by multiplying all of our prime numbers and adding $1$. According to the Fundamental Theorem of Arithmetic, every integer greater than $1$ is either prime or a unique factorization of primes. So let's try both of these possibilities. 
Possibility 1: $a$ is prime. However, we previously wrote all the primes on our complete list, so this is a contradiction. 
Possibility 2: $a$ is composite. However, if it is, it needs to be a unique factorization of the primes on our list. But, it won't divide $P_1$ exactly, $P_2$, $P_3$, or any $P_n$ for that matter. So it is a violation of the Fundamental Theorem of Arithmetic, and therefore a contradiction.
Because we get a contradiction when we assume there are finitely many primes, it must be the opposite, or there must be infinitely many primes. 
Q.E.D
A: The standard way
is by assuming that
there are only a finite number of primes
and deducing that
if all terms of the form
$\prod_{p \in P} p^{a(p)}$
are counted,
there are not enough of them.
I don't remember the details,
but it might go
something like this:
The number of integers
of the form
$\prod_{p \in P} p^{a(p)}$
which are less that $x$
are of the form
$\prod_{p \in P} p^{a(p)}
\le x$
or
$\sum_{p \in P} a(p)\ln p
\le \ln x$.
We certainly have
$a(p) \le \ln x/\ln p$.
The total number of possibilities
is certainly less than,
where $N$ is the number of primes,
$\prod_{p \in P}(\ln x)/(\ln p)
=(\ln x)^{N}\prod_{p \in P}1/(\ln p)
$.
However,
for every finite $N$,
$(\ln x)^{N}$ will be less than $x$
for large enough $x$.
So $N$ must be infinite.
I just did this off
the top of my head,
so I'm not completely sure that
it is correct,
but it seems OK to me.
Anyway,
there it is.
