Context : Me and my friend are taking a basic number theory course for undergrads. I was thinking of Euclid's proof of infinitude of primes and my friend thought of a way to prove it using Pigeon Hole Principle.
Thm : There are infinitely many primes in $\mathbb{N}$.
Proof : Suppose there are finite number of primes and let this number be $m$. Let p be the largest prime. Create m holes and we have p pigeons (i.e primes). Now the natural number $p!+1$ should belong to one of the $m$ holes which isn't the case. So there exists a $(m+1)th$ hole for $p!+1$. This means there are infinitely many primes in $\mathbb{N}$.
I tried to explain to him that this is both redundant and overkill. I wish to know
(1) if my friend is right here
(2) is there any elementary theorems in Number theory in which PHP is used.