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. He/him.

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