COMPLETE AND REDUCED RESIDUE SYSTEMS OF SECOND-ORDER RECURRENCES MODULO p

Fix a prime/?. We say that a set S forms a complete residue system modulop if, for all i such that 0 < i < p 1 , there exists s GS such that s = J (mod /?). We say that a set S forms a reduced residue system modulo p if, for all / such that 1 < i < p -1, there exists s GS such that s = i (mod p). In [9], Shah showed that, ifp is a prime and p = 1,9 (mod 10), then the Fibonacci sequence does not form a complete residue system modulo p. For p > 7, Bruckner [2] proved this result for the remaining case. Thus, ifp is a prime and p > 7, then the Fibonacci sequence {Fn} has an incomplete system of residues modulo/?. Somer [11] generalized these results by considering all linear recurrence sequences with parameters (a, 1), i.e., linear recurrences of the form