Fibonacci Number of Even Index by Golden Mean Modulo 1

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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



Sources