Divisibility Properties of the Fibonacci, Lucas, and Related Sequences

and F n+1 = F n + F n−1 . The first few terms are {1, 1, 2, 3, 5, 8, 13, 21, 34, 55}. The Lucas sequence is a related sequence with the same recurrence L n+1 = L n + L n−1 but different starting values of L 1 = 1 and L 2 = 3. 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 somewell-knowndivisibility 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.


Introduction
The Fibonacci series is one of the most interesting series in mathematics.It is a two-term recurrence, where  1 =  2 = 1 and  +1 =   +  −1 .The first few terms are {1, 1, 2, 3, 5, 8, 13, 21, 34, 55}.The Lucas sequence is a related sequence with the same recurrence  +1 =   +  −1 but different starting values of  1 = 1 and  2 = 3.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  −1 ∈  such that  −1 = 1.
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.
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 gcd(, )) 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 2 ISRN Algebra unique up to multiplication by a unit.Furthermore there exists ,  ∈  such that  +  = gcd(, ).
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 mod ) if and only if gcd(, ) = 1.
Proof.If gcd(, ) = 1, then there exists ,  ∈  such that  +  = 1 which means that  is invertible in /.Conversely, if  is invertible mod , let  be the inverse of  in /.Then  divides  − 1 in  and hence there exists  ∈  such that  +  = 1 and gcd(, ) = 1.
When  = Z, mod  is usually used to denote arithmetic in Z/Z.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].

Divisibility Properties of Fibonacci and Lucas Numbers
In this section, we give matrix theoretical proofs of the wellknown 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  +1  −1 −  2  = (−1)  .We follow the usual convention by letting  denote the matrix ( 1 1 1 0 ).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 ,  ∈ N; then   divides   .
Proof.  is a diagonal matrix mod   which means that   = (  )  is also a diagonal matrix mod   and hence   divides   .Theorem 7.For all ,  ∈ N,  (,) = (  ,   ).
We now derive similar results for the Lucas sequence.We note that if we let  be the matrix ( 1 2 2 −1 ), then   = (  +1      −1 ).We note that  = 2 −  and hence  and  commute.Also note that  2 = 5.Proposition 8. Let ,  ∈ N with  odd; then   divides   .
Proof.We begin by deriving the Cassini identity for the Lucas sequence.By taking determinants, we get  +1  −1 −  2  = det(  ) = (−1) +1 5.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  = 2 + 1.   is a diagonal matrix mod   which means that 5    = (  )  is also a diagonal matrix mod   .Since   is not divisible by five,   divides   .
A nearly identical argument gives us the following result.
We also have a simple proof of the following.
Proof.It follows from Proposition 8 that   divides gcd(  ,   ).We now show that gcd(  ,   ) divides   .If gcd(  ,   ) = 1, we are done so suppose that gcd(  ,   ) > 1.Since no element of the Lucas sequence is divisible by five and det() = −5,  must be invertible mod   for any  ∈ N. 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 mod gcd(  ,   ) and so is (  )  (  )  which is equal to a power of V times   .Hence gcd(  ,   ) divides   and gcd(  ,   ) =   .

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 {  (, )} ∞ =0 is the solution to the recurrence relation  +1 =   −  −1 with initial conditions  0 = 0 and  1 = 1.The generalized Lucas sequences of the second kind {  (, )} ∞ =0 satisfy the exact same recurrence relation  +1 =   −  −1 but have initial conditions of  0 = 2 and  1 = .The discriminant of either kind of generalized Lucas sequences is the quantity Δ =  2 − 4.
The Fibonacci polynomials {  ()} ∞ =1 satisfy the recurrence relation  +1 () =   () +  −1 () with initial polynomials  0 () = 0 and  1 () = 1.The Lucas polynomials {  ()} ∞ =1 satisfy the same recurrence but have different initial polynomials  0 () = 2 and  1 () = .If we relax the condition that  is an integer and allow  to the polynomial , we note that the Fibonacci polynomials are {  (, −1)} ∞ =0 and the Lucas polynomials are {  (, −1)} ∞ =0 .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, {  (, )} ∞ =0 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 (  − 1 0 ).By replacing  with  in Proposition 6, we get the following.
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 Z unless  = ±1.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  +1 is relatively prime to .Therefore, by mathematical induction,   is relatively prime to  for all  ≥ 1 which means that  is invertible mod   for all  ≥ 1 when  and  are relatively prime.Now, by replacing  by  in Theorem 7, we get a proof of the following result.
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  = (  −2 2 − ); then (  +1 −    − −1 ) =   .This identity also follows easily from induction.Taking the determinants of both sides and dividing by −, we get Cassini's identity for these sequences  +1  −1 −  2  = Δ −1 .From this we obtain a useful lemma.
Lemma 14.Let  and  be relatively prime integers with Δ =  2 − 4 ̸ = 0.If  is odd, then Δ must be relatively prime to   for all .If  is even, then Δ is divisible by four; all of the   s are even and   /2 is relatively prime to Δ/4 for all .
Proof.Suppose that  is even.Now we prove the theorem by induction on .The base  = 1 case follows from the fact that  1 /2 = /2 which is relatively prime to Δ/4.The inductive step follows from dividing the Cassini identity by four to get (( −1 /2)( +1 /2)) − (  /2) 2 = (Δ/4) −1 .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  = 2− and hence  and  commute.Also note that  2 = Δ.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 ,  ∈ N with  being odd; then   divides   .