# Fibonacci Number of Even Index by Golden Mean Modulo 1

## Theorem

Let $n \in \Z$ be an integer.

Then:

$F_{2 n} \phi \bmod 1 = 1 - \phi^{-2 n}$
$F_n$ denotes the $n$th Fibonacci number
$\phi$ is the golden mean: $\phi = \dfrac {1 + \sqrt 5} 2$

## Proof

 $\displaystyle F_{2 n} \phi$ $=$ $\displaystyle \phi \dfrac {\phi^{2 n} - \hat \phi^{2 n} } {\sqrt 5}$ Euler-Binet Formula $\displaystyle$ $=$ $\displaystyle \dfrac {\phi^{2 n + 1} - \phi \hat \phi^{2 n} } {\sqrt 5}$ $\displaystyle$ $=$ $\displaystyle \dfrac {\phi^{2 n + 1} + \hat \phi^{2 n - 1} } {\sqrt 5}$ Golden Mean by One Minus Golden Mean equals Minus 1‎