Complete Generalized Fibonacci Sequences Modulo Primes

We study generalized Fibonacci sequences F_n+1=PF_n-QF_n-1 with initial values F_0=0 and F_1=1. Let P,Q be nonzero integers such that P^2-4Q is not a perfect square. We show that if Q=± 1 then the sequence {F_n}_n=0^∞ misses a congruence class modulo every prime large enough. On the other hand, if Q ≠± 1, we prove that (under GRH) the sequence {F_n}_n=0^∞ hits every congruence class modulo infinitely many primes.