Euler indicators of binary recurrence sequences

In this paper, we show that if $(u_n)_n\geq 0$ and $(v_n)_n\geq 0$ are two non-degenerate binary recurrent sequences of integers such that $(v_n)_n\geq 0$ satisfies some technical assumptions, then the diophantine equation $\vert v_n\vert = \phi(\vert u_m\vert)$ has only finitely many effectively computable positive integer solutions $(m,n)$. Here, for a nonzero integer $k$ we use $\phi(k)$ to denote the Euler function of $k$.