On Certain Arithmetic Properties of Fibonacci and Lucas Numbers

mirroring a well-known feature of Fibonacci numbers (see Theorem 2.5). It was pointed out in [1] that (0.2) could itself be used to disprove the corresponding assertion for the 1cm; precisely, if lcm(a, b) = £, then lcm(Ma, Mb) Mt only in the trivial cases a\b or b\a. The argument rested on a uniqueness theorem for the expression of rational numbers as a ratio of two members of the {Mn} sequence. However, the authors did not establish the corresponding negative results for lcm(i^, Fb), lcm (Z,a, Lb). In this paper the gap is filled, precisely by establishing the relevant uniqueness statements for ratios of Fibonacci numbers and Lucas numbers. It turns out that much of the work can be done for arbitrary sequences {un} of positive integers satisfying the recurrence relation un+2 = un+l + un, n>\. Such sequences are, in a sense, classified by their initial values uh u2. However, to discuss the classification, it is better to take the sequences backward with respect to n, that is, to allow n to take any integer value, although the principal results are all to be concerned with positive values of n. Then the Fibonacci sequence {Fn} belongs to the special class given by u0 = 0. Another interesting class, from our point of view, is given by 0 < u0 < ux. The Lucas sequence {Ln} seems, to us, to belong to a singleton class.