Find all positive integers $n$ such that $\phi(\phi(n))=3$.

Question

Find all positive integers $n$ such that $\phi(\phi(n))=3$. Here $\phi(x)$ is Euler's phi-function.

I started by letting $\phi(n)=m$ so essentially the first thing I need to do is find the integers $m$ such that $\phi(m)=3$. However, I know that other than $\phi(x)=1$, there are not other possibilities for $\phi(x)=k$ such that $k$ is odd. Is there something I'm overlooking here such that I would be able to actually find $n$ values that work?

Answer 1

Your reasoning is correct. If $\phi(x)$ cannot equal $3$, then $\phi(\phi(x))$ can't either, because the image of $\phi(\phi(x))$ is a subset of the image of $\phi(x)$