Let $a$ and $b$ be relatively prime integers and let $u_n$ be their associate Lucas sequence, i.e., the second order linear recurrence sequence satisfying $u_0 = 0$, $u_1 = 1$ and $u_{n+2} = au_{n+1} + bu_n$, for each nonnegative integer $n$.

It is well know that $(u_n)_{n=0}^\infty$ is a strong divisibility sequence, i.e., it holds $$(\bullet) \quad \gcd(u_m, u_n) = u_{\gcd(m,n)} ,$$ for all the integers $m,n \geq 0$ (put $\gcd(0,0) := 0$). This in turn implies that $$(\star) \quad m \mid n \Rightarrow u_m \mid u_n ,$$ for all the integers $m,n \geq 0$.

My question is: Are there some nice hypothesis under which also the reverse implication holds in ($\star$) ?

Note that from ($\bullet$), we get $$ u_m \mid u_n \Rightarrow u_m = \gcd(u_m, u_n) = u_{\gcd(m,n)} ,$$ so if $(u_n)_{n=0}^\infty$ is injective then $m = \gcd(m,n)$ and thus $m \mid n$. So my question can be also answered if one gets some nice hypothesis under which $(u_n)_{n=0}^\infty$ is injective.

Thank you in advance for any suggestion.

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