Fibonacci Numbers which equal their Index

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 $4$
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