# Reference requested about the rank of appearance in the Lucas sequences

Given $$u_n$$ a Lucas sequence, we define the rank of appearance of $$m$$ in $$\{u_n\}_{n\geq 0}$$, indicated with $$z_u(m)$$, as the smallest positive integer $$n$$ such that $$m$$ divides $$u_n$$.

I would like to know when $$z_u(m)$$ exists, i.e. what are the conditions on $$\{u_n\}_{n\geq 0}$$ and $$m$$ for which we can find the integer $$z_u(m)$$?

Moreover, what is the general relation between $$z_u(p^{k})$$ and $$z_u(p)$$, for a prime $$p$$ and a natural number $$k\geq 1$$? Then, what are sufficient conditions on $$\{u_n\}_{n\geq 0}$$ and $$m$$ for which such relation holds?

I would be very grateful if someone could answer to my questions explicitely; otherwise, I would be also happy if a good reference was pointed out.