# Golden Ratio Base

The golden ratio $\phi$ is the larger positive root to $x^2 = x + 1$.

We can calculate that

$\begin{array} { l l } \phi^0 = 1, & \\ \phi^1 = \phi, &\phi^{-1} = \phi - 1, \\ \phi^2 = \phi + 1, & \phi^{-2} = -\phi + 2, \\ \phi^3 = 2\phi + 1, & \phi^{-3} = 2\phi -3 . \\ \end{array}$

This allows us to write numbers in base $\phi$, where each place value is a non-negative integer less than $\phi$. For example,
$\begin{array} { l l l l l l } 1 & = & 1 & = & 1 _\phi \\ 2 & = & \phi + (-\phi + 2) & = & 10.01_\phi \\ 3 & = & (\phi+1) + (-\phi + 2) & = & 100.01 _\phi. \end{array}$

Give the finite decimal representation of 5 in base $\phi$ that doesn't use a consecutive pair of 1's.

(You may assume the fact that a finite decimal representation with no consecutive pair of 1's is unique. This is known as the standard form for base $\phi$.)

×