It's a well-known rule that a number $x$ belongs to the Fibonnaci Sequence iff: $$\begin{align}5x^2\pm~4&=\lambda^2&\lambda\in\mathbb Z\end{align}$$ In other words, if and only if $5x^2\pm~4$ is a perfect square.
What is the equivalent identity for testing whether a number is a member of the Lucas sequence?
The equivalent identity for Lucas numbers is: $$5L_n^2\pm~20=\lambda^2$$ as can be found here. To put it another way, a number $x$ is in the Lucas Sequence if and only if $5L_n^2\pm~20$ is a perfect square. Just think $4$ for Fibonacci, $20$ for Lucas.
Matter of fact, it can be shown that $\lambda=5F_n$: $$\begin{align}5F_n^2\pm~4&=L_n^2\\ 5F_n^2&=L_n^2\pm~4\\ 25F_n^2&=5(L_n^2\pm~4)\\ 25F_n^2&=5L_n^2\pm~20\end{align}$$ This is very intriguing!