I have not, as far as I remember, encountered this theorem before. And for the time I’ve had to think about it I realize I’ve got no idea how to prove it. However, it’s a neat little result that makes me smile to hear about, and theorems that bring smiles are certainly worth sharing.

I haven’t written anything here in a while, but hope to write more regularly now that the semester is over—I have a seriesoncombinatorialproofs to finish up, some books to review, and a few other things planned. But to ease back into things, here’s a little puzzle for you. Recall that the Fibonacci numbers are defined by

$latex F_0 = 0; F_1 = 1; F_{n+2} = F_{n+1} + F_n$.

Can you figure out a way to prove the following cute theorem?

If $latex m$ evenly divides $latex n$, then $latex F_m$ evenly divides $latex F_n$.

(Incidentally, the existence of this theorem constitutes good evidence that the “correct” definition of $latex F_0$ is $latex 0$, not $latex 1$.)

For example, $latex 5$ evenly divides $latex 10$, and sure enough, $latex F_5 = 5$ evenly divides $latex F_{10} = 55$. $latex 13$ evenly divides $latex 91$, and sure enough, $latex…

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