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