%0 Journal Article
%T Divisibility Properties of the Fibonacci, Lucas, and Related Sequences
%A Thomas Jeffery
%A Rajesh Pereira
%J ISRN Algebra
%D 2014
%R 10.1155/2014/750325
%X 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
%U http://www.hindawi.com/journals/isrn.algebra/2014/750325/