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.
Charlie Jacobson, Department of Mathematics, Elmira College Chjacobson@elmira.edu
"Unique Properties of the Fibonacci and Lucas Sequences,"
Rose-Hulman Undergraduate Mathematics Journal: Vol. 9
, Article 5.
Available at: http://scholar.rose-hulman.edu/rhumj/vol9/iss1/5