Long Days on the Fibonacci Clock

Abstract

The rule generating the sequence is extraordinarily simple, but its repeated application produces rich mathematics. If we leave the rule alone and change the start ing values, we again find sequences with interesting properties?such as the Lucas If we introduce modular arithmetic, there are new questions to answer. In what follows, I study sequences in ?p = Z/pZ, the integers mod p, where p is a prime. I like to call these the Fibonacci numbers. Regardless of the start ing pair, the sequence will repeat [10]. The question I want to answer is What is the maximum period for any sequence in Z/pZT In what follows, I present a particular point of view about the sequence in a way that gives some insight into both the standard sequence and its variations. Specifically, the sequence is interpreted in terms of a matrix acting on a finite set, an idea that is related to group actions and to (discrete) dynamical systems. The underlying set is the two-dimensional vector space ?p2; the matrix M from (2) provides the rule for the process. Iterations of the system correspond to powers of the matrix. Periods in the system are related, then, to powers of M that are equivalent modulo p to the identity matrix. The point of view works for all cases, even for the generalized numbers, where weights are allowed in the recursion formula. For a thorough look at dynamical systems and number theory, Silverman's book [8] is an excellent source. Most of what is contained here is not new. Searching Mathematical Reviews turns up dozens of articles about periods of numbers in Z/mZ, including many where m doesn't even have to be a prime or a power of a prime. The most-referenced article is Wall's article [10] in the monthly in 1960. Wall established many funda mental results, and posed some tantalizing questions. In particular, he showed that the period divides (p ? 1) when 5 is a quadratic residue mod p and divides (2/7 + 2) when it is not, but he did not find the maximal periods. Wall's investigation was motivated by a search for methods of generating pseudorandom Later, Brent [1], also motivated by pseudorandom numbers, considered the special properties of sequences modulo a power of 2. The story, however, begins even before the days of Mathematical Reviews. In the 1930s, Ward [11] considered periods, both minimal and maximal, and other characteristics of sequences arising from rather general recurrence relations, not just the relation. Kaiman and Mena's article in an earlier issue of this Magazine [5] examines many of the famous properties of the num bers as specific instances of properties of general second-order recurrences. Ward's results built on even earlier work by Carmichael [2] and others. For the early history of the subject, the curious reader should consult Dickson's history [3], particularly Volume I, Chapter XVII, where elements of the problem are traced back to Gauss and Lagrange. Earlier in the Magazine, Vella and Vella [9] looked at possible periods in