On the Divisibility of Fibonacci Sequences by Primes of Index Two

Brother Alfred has characterized primes dividing every Fibonacci sequence [2] based on their period and congruence class mod 20. More recently, in [4] Ballot and Elia have described the set of primes dividing the Lucas sequence, meaning they divide some term of the sequence. Our purpose here is to extend the results of the former paper utilizing the methods of the latter. In particular we will investigate primes dividing “half of all Fibonacci sequences,” defined in our case via their index rather than period. We will establish a simple criterion for determining whether or not such a prime divides a given Fibonacci sequence based on whether or not the norm of that sequence is a quadratic residue modulo the prime. The discussion is organized as follows. We begin by giving a concise development of the Wythoff array, in which we adopt a novel approach and provide new proofs of wellknown properties. In the following section we define the norm of a Fibonacci sequence, so-named because this value is given by the algebraic norm of the element of the ring of integers Z[φ] used to generate that sequence. After presenting several properties of the norm we define the index of a prime p relative to the Fibonacci sequence and relate this notion to the multiplicative index of the element −φ modulo a prime over p in Z[φ]. We next employ these ideas to prove the criterion for when a Fibonacci sequence is divisible by a prime of index 2, then conclude with several observations and examples. Finally, before beginning we acknowledge the many results that have already been found regarding divisibility of Fibonacci sequences by primes, as presented in [5], [6], [8], [11], and [12], for instance.