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.

I have searched a bit on the internet, but I have not found a complete answer.

Thank you very much for your help!


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