Divisibility Properties of the Fibonacci, Lucas, and Related Sequences

Abstract

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.

1. Introduction

The Fibonacci series is one of the most interesting series in mathematics. It is a two-term recurrence, where and . The first few terms are . The Lucas sequence is a related sequence with the same recurrence but different starting values of and . The Fibonacci and Lucas sequences are special cases of the generalized Lucas sequences studied by Lucas in [1]. We will study these sequences in section two and the Gaussian Fibonacci sequences of Jordan [2] will be studied in section three. In this paper, we will give some easy matrix theoretical proofs of some well-known divisibility properties of these sequences. All of these proofs use the arithmetic of matrices over rings and two elementary ideas: Bezout’s identity and the fact that any power of a diagonal matrix is a diagonal matrix. This gives an elementary and unified derivation of the divisibility properties of all of these sequences. We begin by reviewing some of the elementary terminologies of rings and properties of matrices over rings. We only assume that the reader is familiar with the definition of a principal ideal domain.

Definition 1. Let be a commutative ring with identity and let . Then is called a unit of if there exists such that .

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.

Proposition 2. Let be a commutative ring with identity and let be a two by two matrix with entries in . If the determinant of is a unit, then is an invertible matrix.

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.

Proof. If is invertible, then simple matrix multiplication shows us that .

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

Definition 3. Let be a principal ideal domain and let ; then an element is called the greatest common divisor of and (denoted by ) if is a divisor of both and and if any other common divisors of both and also divide .

Proposition 4. Let be a principal ideal domain and let . Then the greatest common divisor of and exists and is unique up to multiplication by a unit. Furthermore there exists such that .

Proof. Let be the ideal generated by and . Then . As is a principal ideal domain, is generated by a single element. This element is unique up to multiplication by a unit. Clearly the generator of satisfies both properties of the GCD and any GCD of and will be a generator of .

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.

Proposition 5. Let be a principal ideal domain and let . Do not let be a unit. Then is invertible in (or equivalently is invertible ) if and only if .

Proof. If , then there exists such that which means that is invertible in . Conversely, if is invertible mod , let be the inverse of in . Then divides in and hence there exists such that and .

When ,    is usually used to denote arithmetic in . We will also use the mod for arithmetic in when is a general principal ideal domain. These are all the results in linear algebra and ring theory that we will need. The further theory of matrices over principal ideal domains as well as many other interesting topics in matrix theory can be found in [3].

2. Divisibility Properties of Fibonacci and Lucas Numbers

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 which can easily be proven by induction. This identity forms the basis for one of the standard proofs of Cassini’s identity . We follow the usual convention by letting denote the matrix . The matrix has appeared in many proofs; see [4] for a detailed history of the matrix. We note that is an invertible matrix. As a demonstration of our methods, we provide a one-line proof of the following well-known divisibility property of the Fibonacci sequence.

Proposition 6. Let ; then divides .

Proof. is a diagonal matrix mod which means that is also a diagonal matrix mod and hence divides .

Theorem 7. For all , .

Proof. It follows immediately from Proposition 6 that divides . We now show that divides . If , we are done so suppose that . Let and be integers such that . Then (one of or will be negative which is not a problem since is invertible and the inverse of a diagonal matrix is diagonal). Since and are diagonal mod , so is . Hence divides which means that .

We now derive similar results for the Lucas sequence. We note that if we let be the matrix , then . We note that and hence and commute. Also note that .

Proposition 8. Let with odd; then divides .

Proof. We begin by deriving the Cassini identity for the Lucas sequence. By taking determinants, we get . It follows that no element of the Lucas sequence is divisible by five as this would force all elements of the Lucas sequence to be divisible by five. Since is odd, let . is a diagonal matrix mod which means that is also a diagonal matrix mod . Since is not divisible by five, divides .

A nearly identical argument gives us the following result.

Proposition 9. Let with even; then divides .

We also have a simple proof of the following.

Theorem 10. Let and let . If and are both odd, then .

Proof. It follows from Proposition 8 that divides . We now show that divides . If , we are done so suppose that . Since no element of the Lucas sequence is divisible by five and , must be invertible mod for any . Let and be integers such that . We note that one of to must be odd and the other must be even. and are both diagonal matrices and so is which is equal to a power of times . Hence divides and .

3. Generalized Lucas Sequences

In [1], Edouard Lucas investigated some useful sequences which have come to be known as the generalized Lucas sequences. We will show that the matrix methods of the previous section can be used to give some simple proofs of the divisibility properties of the generalized Lucas sequences. These divisibility properties can be found in Lucas’ original paper [1] (see also [5] or chapter 1 of [6]).

Definition 11. Let and be integers; then the generalized Lucas sequence of the first kind is the solution to the recurrence relation with initial conditions and . The generalized Lucas sequences of the second kind satisfy the exact same recurrence relation but have initial conditions of and . The discriminant of either kind of generalized Lucas sequences is the quantity .

We are using lower case and instead of the more standard capital and so as to avoid confusion with the matrix. Many important integer sequences are a special case of the generalized Lucas sequence. Consider that are the Fibonacci numbers and are the Lucas sequence. Consider that are the Pell numbers; consider that are the Mersenne numbers and are the Jacobsthal numbers and are the Jacobsthal-Lucas numbers. In all of these cases and are relatively prime and is nonzero. We note that the only choices of and which are relatively prime and for which are and . It can be verified that , , , and ; the divisibility properties of these sequences are obvious and hence we are justified in restricting ourselves to the cases where is nonzero.

The Fibonacci polynomials satisfy the recurrence relation with initial polynomials and . The Lucas polynomials satisfy the same recurrence but have different initial polynomials and . If we relax the condition that is an integer and allow to the polynomial , we note that the Fibonacci polynomials are and the Lucas polynomials are . Our methods in this section also apply to these polynomials and hence the conclusions of Proposition 12 and Theorem 13 apply also to the Fibonacci polynomials and the conclusions of Proposition 15 and Theorem 16 also apply to the Lucas polynomials.

We will show that, if and are relatively prime, satisfies the same divisibility properties as the Fibonacci numbers. We will use the matrix identity which can easily be proven by induction. This identity first appears in [7]. In the remainder of this section, we let denote .

By replacing with in Proposition 6, we get the following.

Proposition 12. Let ; then divides .

Proof. is a diagonal matrix mod which means that is also a diagonal matrix mod and hence divides .

We can also prove a generalization of Theorem 7. unlike will not be invertible over unless . However will be invertible mod if and are relatively prime. We show this using the fact that will be invertible mod if and only if and are relatively prime. It follows immediately from the difference equation that if is relatively prime to , then is relatively prime to . Therefore, by mathematical induction, is relatively prime to for all which means that is invertible mod for all when and are relatively prime. Now, by replacing by in Theorem 7, we get a proof of the following result.

Theorem 13. Let and be relatively prime integers. Then for all , .

Proof. It follows immediately from Proposition 12 that divides . We now show that divides . If , we are done so suppose that . Let and be integers such that . Then is diagonal . Since and are diagonal mod , so is . Hence divides which means that .

We note that our proofs of Proposition 12 and Theorem 13 also work for the Fibonacci polynomials.

We also have a matrix identity for the generalized Lucas sequences of the second kind. Let ; then . This identity also follows easily from induction. Taking the determinants of both sides and dividing by , we get Cassini’s identity for these sequences . From this we obtain a useful lemma.

Lemma 14. Let and be relatively prime integers with . If is odd, then must be relatively prime to for all . If is even, then is divisible by four; all of the s are even and is relatively prime to for all .

Proof. Suppose that is even. Now we prove the theorem by induction on . The base case follows from the fact that which is relatively prime to . The inductive step follows from dividing the Cassini identity by four to get . The case where is odd is proved similarly.

Note that is relatively prime to for all which means that is invertible mod for all (if shares a prime factor with one element of the sequence , then by Cassini’s identity every element of this sequence would be divisible by this factor including ). We note that and hence and commute. Also note that . We can now prove the following by replacing with and with in the proof of Proposition 8.

Proposition 15. Suppose that and are relatively prime integers. Let with being odd; then divides .

Proof. If then we are done, so suppose that . Since is odd, let . Notice that is a diagonal matrix which means that is also a diagonal matrix mod . If is odd, then is relatively prime to , which means that divides . If is even, then is a diagonal matrix mod which means that is also a diagonal matrix mod . Since is relatively prime to , divides .

By replacing with and with in the proof of Theorem 10, we have a simple proof of the following.

Theorem 16. Suppose that and are relatively prime integers and . Let . If and are both odd, then .

Proof. If , then we must have and for and to be relatively prime integers. In this case for all and our result clearly holds. We now suppose that is nonzero which means that the matrix is invertible. It follows from Proposition 15 that divides . We now show that divides . If , we are done, so suppose . Let and be integers such that . We note that one of must be odd and the other must be even. Suppose that is odd; then and are both diagonal matrices mod and so is which is equal to a power of times . Hence divides and . Now suppose that is even; then both and are diagonal matrices mod . Since and is relatively prime to for all , is diagonal mod . Hence divides and .

We can also use matrix techniques to prove some relations between the and the for fixed values of and .

Lemma 17. Let and be integers and let . Then .

Proof. We can easily verify that the result holds for and . Since the trace of is and the determinant of is , has characteristic polynomial . By the Cayley-Hamilton theorem, for all ; our result follows.

By taking the determinants of both sides of , we get the following identity.

Corollary 18. Let and be integers and let ; then .

4. Gaussian Fibonacci Numbers

All of the divisibility results in this section are new proofs of results in [2] where the Gaussian Fibonacci numbers were first defined. The Gaussian Fibonacci numbers are a sequence of Gaussian integers. The ring of Gaussian integers is the set of complex numbers whose real and imaginary parts are both integers. Consider that . The Gaussian integers form a unique factorization domain.

We follow [2] in defining the Gaussian Fibonacci numbers as follows. Let , , and . It follows from induction that . We now let . We note that and . By taking determinants of both sides of the latter equation, we get Cassini’s formula for the Gaussian Fibonacci numbers . We note that is a Gaussian prime which divides the right-hand side of Cassini’s formula and hence it is not a factor of any of the Gaussian Fibonacci numbers (if it were, it would be a factor of all of them including , which is not the case). Since , is invertible for all . It also follows from these two equations that for all . We can now use these results to give new proofs of the divisibility properties of the Gaussian Fibonacci numbers.

Proposition 19. Let . If divides , then divides .

Proof. is a diagonal matrix mod . Therefore if divides , then is also a diagonal matrix mod and hence divides .

Theorem 20. Let and let be the natural number defined such that ; then .

Proof. It follows immediately from Proposition 19 that divides . We now show that divides . If , we are done, so suppose that . Let and be integers such that . Then . Since and are diagonal mod , so is . Hence divides . Since does not divide any Gaussian Fibonacci number, divides which means that .

We note that we can introduce the following sequence with a parameter which contains as special cases the Fibonacci, Lucas, and the Gaussian Fibonacci sequences: . When we get the ordinary Fibonacci sequence, when we get the Lucas sequence, and when we get the Gaussian Fibonacci sequence. We note that . Taking the determinant of both sides, we get which gives us the Cassini identity for all three of the sequences at once. If we take to be either or , the right-hand side of the previous equation becomes zero and hence and are both geometric sequences. We note that . The values of which make either the term or the term on the right-hand side of this equation disappear are exactly the choices of which give us the Fibonacci sequence, the Lucas sequence, the Gaussian Fibonacci sequence, or the conjugate of the Gaussian Fibonacci sequence. This goes some way towards explaining why these sequences amongst those we get for other choices of have such nice divisibility properties.

Conflict of Interests

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

Acknowledgment

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