A period 5 difference equation


Department of Math. Sci., Appalachian State University, Boone, NC 28608, USA


The main goal of this note is to introduce another second-order difference equation where every nontrivial solution is of minimal period 5, namely the difference equation:
$$x_{n+1} =\frac{1 + x_{n−1}}{x_nx_{n−1} − 1},      n = 1, 2, 3, . . .$$
with initial conditions $x_0$ and $x_1$ any real numbers such that $x_0x_1 \neq 1$.