I am trying to solve a question in Computabiltiy Theory which says:
Let $f, g$ be two computable functions. Prove that there exist numbers $e_1, e_2$ such that:
$ \phi_{f(e_1)} = \phi_{e_2}$
$ \phi_{g(e_2)} = \phi_{e_1}$
Where $\phi_n$ denotes the program with the corresponding code $n$.
My original idea was to use the Recursion Theorem and construct a computable function that's able to work with the fixed points of the two functions $f,g$ and construct the $e_1, e_2$ that way. But that seems to not work and I am not sure if I'm on the right path or not.
I would appreciate any hints or ideas on the problem.