If we have that $p\mid u_a$, $b\mid a$, and $p\mid u_a/u_b$, prove that $p\mid n/b$, assuming that $u_a$ and $u_b$ are terms in the linear recurrence for the Lucas Sequence.
I've tried looking at the characteristic polynomial and using the discriminant in divisibility comparisons, but I haven't be able to get that far. This is one idea of where to go:
$u_n/u_m = (a^n-b^n)/(a^m-b^m) = K_{n/m}$
The characteristic polynomial of $K_{n/m}$ is $(a^m, b^m)^2$, but that's as far as I know.
If anyone has ideas for how to approach the proof, it would be much appreciated.