Long-Term Behavior of Solutions of the Difference Equation xn+1=xn-1xn-2-1

and Applied Analysis 3 Proof. Assume that xN xN 2k and xN 1 xN 2k 1, for every k ∈ N0, and some N ≥ −2, with xN / xN 1. Then, we have xN 4 xN 2xN 1 − 1 xNxN 1 − 1 xN 3 xN 1. 2.4 From this and since xN 4 xN , we obtain a contradiction, finishing the proof of the result. Theorem 2.3. There are no periodic or eventually periodic solutions of 1.1 with prime period three. Proof. If xN xN 3k, xN 1 xN 3k 1, xN 2 xN 3k 2, k ≥ 0, 2.5 for some N ≥ −2, we have xN 3 xN 1xN − 1 xN, xN 4 xN 2xN 1 − 1 xN 1, xN 5 xN 3xN 2 − 1 xNxN 2 − 1 xN 2. 2.6 If xN 0, xN 1 0, or xN 2 0, then from 2.6 we easily obtain contradictions in all these cases. Hence, wemay assume that xN / 0, xN 1 / 0, and xN 2 / 0. Equalities in 2.6 also imply that xN 1 xN 1 xN , xN 2 xN 1 1 xN 1 , xN xN 2 1 xN 2 . 2.7