Fibonacci and Lucas Numbers with Applications, Volume 1, 2nd Edition

DIVISIBILITY PROPERTIES

In this chapter, we will explore a few divisibility properties of Fibonacci and Lucas numbers.

10.1 FIBONACCI DIVISIBILITY

In Chapter 5, we learned that , so . Can we generalize this? In other words, under what conditions does ? The next theorem gives us a partial answer. We will establish it using PMI.

Theorem 10.1

Let . Then .

Proof

Clearly the given statement is true when .

Now assume it is true for all positive integers , where . We will now show that