The Order of the Fibonacci and Lucas Numbers

In this paper vp(r) denotes the exponent of the highest power of a prime p which divides r and is referred to as the/?-adic order of r. We characterize the/?-adic orders vp(F„) and vp(Ln), i.e., the exponents of a prime/? in the prime power decomposition of Fn and Ln, respectively. The characterization of the divisibility properties of combinatorial quantities has always been a popular area of research. In particular, finding the highest powers of primes which divide these numbers (e.g., factorials, binomial coefficients [14], Stirling numbers [2], [1], [10], [9]) has attracted considerable attention. The analysis of the periodicity modulo any integer (e.g., [3], [11], [14], [8]) of these numbers helps exploring their divisibility properties (e.g., [9]). The periodic property of the Fibonacci and Lucas numbers has been extensively studied (e.g., [16], [13], [17], [12]). Here we use some of these properties and methods to find vp(Fn) and vp(Ln). An application of the results to the Stirling numbers of the second kind is discussed at the end of the paper. We note that Halton [5] obtained similar results on the /?-adic order of the Fibonacci numbers, and additional references on earlier developments can be found in Robinson [13] and Vinson [15]. The approach presented here is based on a refined analysis of the periodic structure of the Fibonacci numbers by exploring its properties, in particular, around the points where Fn = 0 (mod /?). [The smallest n such that Fn = 0 (mod /?) is called the rank of apparition of prime p and is denoted by n(p).] This technique is based on that of Wilcox [17] and provides a simple and selfcontained analysis of properties related to divisibility. For instance, we obtain another characterization of the ratio of the period to the rank of apparition [15] in terms of Fn^pyX (mod/?) for any prime/?.