When does the unit group of $\Bbb Z_n$ have prime order?

Question

There are cases when the multiplicative group $\mathbb{Z}_n^*$ is a cyclic group. But my question is, for what $n$ does $\mathbb{Z}_n^*$ have prime order?

Or equivalently, for what $n$ is the Euler totient function of $n$ equal to a prime number?

Answer 1

$\phi(2^k)=2^{k-1}$ is even for $k>1$,

$\phi(p^k)=(p-1)p^{k-1}$ is even for odd prime $p$ and $k\ge1$,

and $\phi$ is multiplicative, so $\phi(n)$ is even for $n>2$.

Therefore, $\phi(n)$ is $2$ or a multiple of $2$ (not prime) for $n>2$.

$\phi(n)=2$ only for $n=3, 4$ or $6$.