We use matrix techniques to give simple proofs of known divisibility properties of the Fibonacci, Lucas, generalized Lucas, and Gaussian Fibonacci numbers. Our derivations use the fact that products of diagonal matrices are diagonal together with Bezout’s identity.

The Fibonacci series is one of the most interesting series in mathematics. It is a two-term recurrence, where

Let

We will use two by two matrices over certain rings to give some easy proofs of some of the divisibility properties of these sequences. We will need the following result.

Let

In fact, the converse of this result is true as well and both of the original proof and the converse remain true for square matrices of arbitrary size.

If

We now introduce the concept of the greatest common divisor and note some of its properties.

Let

Let

Let

This result is sometimes called Bezout’s identity. We can use Bezout’s identity to prove the following result which will be useful later on.

Let

If

When

In this section, we give matrix theoretical proofs of the well-known divisibility properties of the Fibonacci and Lucas numbers. Our proofs in this section use the well-known fact that

Let

For all

It follows immediately from Proposition

We now derive similar results for the Lucas sequence. We note that if we let

Let

We begin by deriving the Cassini identity for the Lucas sequence. By taking determinants, we get

A nearly identical argument gives us the following result.

Let

We also have a simple proof of the following.

Let

It follows from Proposition

In [

Let

We are using lower case

The Fibonacci polynomials

We will show that, if

By replacing

Let

We can also prove a generalization of Theorem

Let

It follows immediately from Proposition

We note that our proofs of Proposition

We also have a matrix identity for the generalized Lucas sequences of the second kind. Let

Let

Suppose that

Note that

Suppose that

If

By replacing

Suppose that

If

We can also use matrix techniques to prove some relations between the

Let

We can easily verify that the result holds for

By taking the determinants of both sides of

Let

All of the divisibility results in this section are new proofs of results in [

We follow [

Let

Let

It follows immediately from Proposition

We note that we can introduce the following sequence with a parameter

The authors declare that there is no conflict of interests regarding the publication of this paper.

The research was supported by the Natural Sciences and Engineering Research Council of Canada Discovery Grant no. 400550.