# 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

- 1997: Donald E. Knuth:
*The Art of Computer Programming: Volume 1: Fundamental Algorithms*(3rd ed.) ... (previous) ... (next): $\S 1.2.8$: Fibonacci Numbers: Exercise $31$