Complete Fibonacci Sequences in Finite Fields

which evidently has the required property. In [1], a similar example is given for]F109* It is implicit in [1], [12], that such sequences exist in 3Fp if ~Fp contains a so-called Fibonacci Primitive Root, or FPR: see below for definitions. Here we show (Theorem 4.2) that such sequences exist in Wp if and only if Wp contains an FPR; moreover, when Wp does contain an FPR, we show that the only such sequences to exist are the "natural' ones: that is, the sequences of successive powers of FPRs. Of course, it was shown in [1] that if the sequence of successive powers of an element is to have this Fibonacci property, then the element in question must be an FPR, but here we allow for any sequence of elements. We also prove (Theorem 4.4) analogous results for Fibonacci-type sequences of the set of (nonzero) squares of ¥p. In this context, the sequence