Some Divisibility Properties of Generalized Fibonacci Sequences

Let c be any square-free integer, p any odd prime such that (c/p) = -1, and n any positive integer. The quantity ./IT, which would ordinarily be defined (mod p) as one of the two solutions of the congruence: x E c (mod p n ) , does not exist. Nevertheless, we may deal with objects of the form a + b/c~(mod p), where a and b are integers, in much the same way that we deal with complex numbers, the essential difference being that /^Ts role is assumed by /~c~. Since we are dealing with congruences (mod p), we may without loss of generality restrict a and b to a particular residue class (mod p), the most convenient for our purpose being the minimal residue class (mod p). Accordingly, we define the sets Rn(p) and Rn(p9c) as follows: