Fibonacci Numbers which equal their Index
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Theorem
The only Fibonacci numbers which equal their index are:
| \(\ds F_0\) | \(=\) | \(\ds 0\) | ||||||||||||
| \(\ds F_1\) | \(=\) | \(\ds 1\) | ||||||||||||
| \(\ds F_5\) | \(=\) | \(\ds 5\) |
Proof
By definition of the Fibonacci numbers:
| \(\ds F_0\) | \(=\) | \(\ds 0\) | ||||||||||||
| \(\ds F_1\) | \(=\) | \(\ds 1\) |
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 $4$