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$\newcommand{\lcm}{\operatorname{lcm}}$There is a nice property of Fibonacci numbers which says that:

$$\gcd(F_{a_1}, \ldots, F_{a_n}) = F_{\gcd(a_1, \ldots, a_n)}$$

I am curious is there anything similar with respect to $\lcm$?

Clearly $\lcm(F_{a_1}, \ldots, F_{a_n}) = F_{\lcm(a_1,\ldots, a_n)}$ is not true, but is there anything better then to use $\lcm(a, b) = \frac{ab}{\gcd(a, b)}$ and to use associative law of $\lcm$?

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  • $\begingroup$ Ad Infinitum 10 has published the editorial~~~ $\endgroup$ – atupal Feb 23 '15 at 12:32
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The problem is already studied here (see Corollary 3.4). It is shown that the least common multiple of two Fibonacci numbers is a Fibonacci number if and only if one of them divides the other.

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