I am trying to prove that $\phi(1)=1$ is the only case where $\phi (n)=n$
I can obviously discount all prime numbers due to the fact that $\phi(p) = (p-1)$
Whenever I try and prove the rest of the cases (for example, using $\phi(mn)=\phi(m)\phi(n)$), I keep falling into an endless loop of needing to prove that the only case where $\phi(n)=n$ only holds for 1.
$1$ is the only number relatively prime to itself. Any other $n>1$ will have a maximum $n-1$ numbers from $1$ to $n$ itself that can be relatively prime to it, having to rule out at least $n$