Find all the primes s.t.: $\phi\left(q^3\right)-2\phi\left(q^2\right)=q+3$

Question

Find all the primes $q$ s.t.:

$$\phi\left(q^3\right)-2\phi\left(q^2\right)=q+3$$

My attempt:

$$\phi\left(q^3\right)-2\phi\left(q^2\right)=q+3$$ $$q^3-q^2-2q^2+2q=q+3$$ $$q^3-3q^2+q-3=0$$ $$(q-3)(q^2+1)=0$$ $\implies$ $q=3$ is the only prime solution.

Is this correct?

Answer 1

Yes, that’s correct. $ $