Relation between Square of Fibonacci Number and Square of Lucas Number

Theorem
Let $F_n$ denote the $n$th Fibonacci number.

Let $L_n$ denote the $n$th Lucas number.

Then:
 * $5 {F_n}^2 + 4 \paren {-1}^n = {L_n}^2$

Proof
Let:
 * $\phi = \dfrac {1 + \sqrt 5} 2$
 * $\hat \phi = \dfrac {1 - \sqrt 5} 2$

Note that we have:

Then:

Hence the result.