Multiplicities of second order linear recurrences

Abstract:

A second order linear recurrence is a sequence $\{ {a_n}\}$ of integers satisfying a ${a_{n + 2}} = M{a_{n + 1}} - N{a_n}$ where N and M are fixed integers and at least one ${a_n}$ is nonzero. If k is an integer, then the number $m(k)$ of solutions of ${a_n} = k$ is at most 3 (respectively 4) if ${M^2} - 4N < 0$ and there is an odd prime $q \ne 3$ (respectively q = 3) such that $q|M$ and $q\nmid kN$. Further $M = {\sup _k}{\;_{{\text {integer}}}}m(k)$ is either infinite or $\leq 5$ provided that either (i) $(M,N) = 1$ or (ii) $6\nmid N$.