On Certain Arithmetic Properties of Fibonacci and Lucas Numbers
نویسندگان
چکیده
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.
منابع مشابه
A Probabilistic Primality test Based on the Properties of Certain Generalized Lucas Numbers
After defining a class of generalized Fibonacci numbers and Lucas numbers, we characterize the Fibonacci pseudoprimes of the mth kind. In virtue of the apparent paucity of the composite numbers which are Fibonacci pseudoprimes of the mth kind for distinct values o f the integral parameter m , a method, which we believe to be new, for finding large probable primes is proposed. An efficient compu...
متن کاملOn the Fibonacci and Lucas p-numbers, their sums, families of bipartite graphs and permanents of certain matrices
In this paper we consider certain generalizations of the well-known Fibonacci and Lucas numbers, the generalized Fibonacci and Lucas p-numbers. We give relationships between the generalized Fibonacci p-numbers, Fp(n), and their sums, Pn i1⁄41F pðiÞ, and the 1-factors of a class of bipartite graphs. Further we determine certain matrices whose permanents generate the Lucas p-numbers and their sum...
متن کاملTrigonometric Expressions for Fibonacci and Lucas Numbers
The amount of literature bears witness to the ubiquity of the Fibonacci numbers and the Lucas numbers. Not only these numbers are popular in expository literature because of their beautiful properties, but also the fact that they ‘occur in nature’ adds to their fascination. Our purpose is to use a certain polynomial identity to express these numbers in terms of trigonometric functions. It is in...
متن کاملThe 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 ind...
متن کاملOn the Properties of Balancing and Lucas-Balancing $p$-Numbers
The main goal of this paper is to develop a new generalization of balancing and Lucas-balancing sequences namely balancing and Lucas-balancing $p$-numbers and derive several identities related to them. Some combinatorial forms of these numbers are also presented.
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
دوره شماره
صفحات -
تاریخ انتشار 2010