Fibonacci Number is not Product of Two Smaller Fibonacci Numbers

Theorem
Let $m, n \in \Z$ be integers.

Suppose $\size m, \size n \ge 3$.

Let $F_m$ and $F_n$ be the $m$th and $n$th Fibonacci numbers.

Then $F_m \times F_n$ is not a Fibonacci number.

Proof
From Honsberger's Identity:
 * $F_n = F_{k - 1} F_{n - k + 2} + F_{k - 2} F_{n - k + 1}$

for $2 \le k \le n$.

$F_n = F_m F_k$ for some $m, k \ge 3$.

Then:

The is a weighted mean of $2$ consecutive Fibonacci numbers.

Thus:
 * $F_{n - k + 2} < F_m < F_{n - k + 1}$

which cannot happen.

The result follows by Proof by Contradiction.