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.

The Math Less Traveled

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…

View original post 47 more words


Author: Joseph Nebus

I was born 198 years to the day after Johnny Appleseed. The differences between us do not end there.

Please Write Something Good

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s