Cycles in the Generalized Fibonacci Sequence modulo a Prime

Since their invention in the thirteenth century, Fibonacci sequences have intrigued mathematicians. As well as modeling the population patterns of overly energetic rabbits, however, they have sparked developments in more serious mathematics. For example, generalized Fibonacci sequences crop up in all manner of situations, from fiber optic networks [3] to computer algorithms [1] to probability theory [2]. In this article, we study generalized Fibonacci sequences {G(n)}, given by the recurrence relation: G(n) = aG(n − 1)+ bG(n − 2) for a, b, G(0) and G(1) integers. We also study the periods of repetition in such sequences when considered modulo p, a prime. For one particular class of generalized Fibonacci numbers, we find a surprising connection with Fermat’s Last Theorem. Other connections between these two seemingly unrelated subjects have been discovered in the past [8], but the one unearthed here allows us to calculate the length of these repetitions or cycles exactly.