Fibonacci Numbers which equal the Square of their Index

Theorem

The only Fibonacci numbers which equal the square of their index are:

\(\ds F_0\)

\(=\)

\(\ds 0\)

\(\ds F_1\)

\(=\)

\(\ds 1\)

\(\ds F_{12}\)

\(=\)

\(\ds 12^2 = 144\)

Proof

By definition of the Fibonacci numbers:

\(\ds F_0\)

\(=\)

\(\ds 0\)

\(\ds F_1\)

\(=\)

\(\ds 1\)

Then it is observed that $F_{12} = 144$.

After that, for $n > 12$, we have that $F_n > n^2$.

$\blacksquare$

Also see

• Square Fibonacci Number: after $144$, there are no more square Fibonacci numbers at all.

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 $5$