The algebraic structure of the set of all Fibonacci-like sequences, which includes the Fibonacci and Lucas sequences, is developed, utilizing an isomorphism between this set and a subset of the 2 by 2 integer matrices. Using this isomorphism, determinants of sequences, and Fibonacci-like matrices, can be defined. The following results are then obtained: (1) the Fibonacci sequence is the only such sequence with determinant equal to 1, (2) the set of all Fibonacci-like sequences forms an integral domain, (3) even powers of Lucas matrices are multiples of a Fibonacci matrix, and (4) only powers of multiples of Fibonacci matrices or Lucas matrices are multiples of Fibonacci matrices.

Author Bio

Stephen Parry will graduate with his BA in Mathematics from Elmira College in 2008, after which he will pursue his PhD in Mathematics. Unique Properties of the Fibonacci and Lucas Sequences was completed under a privately funded research grant at Elmira College advised by Dr. Charlie Jacobson. Stephen is interested in algebra, topology, sushi, Latex, and photography.