# Fibonacci Numbers which equal their Index

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## Theorem

The only Fibonacci numbers which equal their index are:

\(\displaystyle F_0\) | \(=\) | \(\displaystyle 0\) | |||||||||||

\(\displaystyle F_1\) | \(=\) | \(\displaystyle 1\) | |||||||||||

\(\displaystyle F_5\) | \(=\) | \(\displaystyle 5\) |

## Proof

By definition of the Fibonacci numbers:

\(\displaystyle F_0\) | \(=\) | \(\displaystyle 0\) | |||||||||||

\(\displaystyle F_1\) | \(=\) | \(\displaystyle 1\) |

Then it is observed that $F_5 = 5$.

After that, for $n > 5$, we have that $F_n > n$.

$\blacksquare$

## 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 $4$