The Number Field Q(/5) and the Fibonacci Numbers

where 0) = %(1 + V5) . It is well known that Z(OJ) is a Euclidean domain [6, pp. 214-15], and that the units of Z(oo) are given by ±0), where nEZ [6, p. 221]. The Binet formula _ _ Fn = (00 03)/((A) W) = (0D 0))/>/5, where 0) = %(1 v5) is the conjugate of 0), expresses the n Fibonacci number in terms of the unit 0). Simiarly, the n Lucas number is given by Ln = b) + 0)". Also, an elementary induction argument using the result (i) = 03 + 1 shows that 0) = Fn,1 + Fnb) for n > 1. These results suggest that the arithmetic theory of Z(oo) can be a powerful tool in the investigation of the arithmetical properties of the Fibonacci and Lucas numbers. This is indeed the case, and the articles by Carlitz [4],Lind [10], and Lagarias & Weisser [9] utilize Z(oo) on a limited scale. In this paper, I further document the utility of Z(OJ) by deriving many of the familiar divisibility properties of the Fibonacci numbers using the arithmetic theory of Z((JO). Much of the development has been adapted from pages 164-174 of my doctoral dissertation [5], which gives a comprehensive treatment of number theory in Z(co).