On the periodic nature of the solutions of the reciprocal difference equation with maximum (Q1876728)

The author investigates the periodic nature of the positive solutions of the difference equation \[ x_n=\max\bigg\{\frac{A}{x_{n-1}},\frac{B}{x_{n-2}},\frac{C}{x_{n-3}}\bigg\}, \;n=0,1,\dots,\eqno{(E)} \] where \(A,B,C\) are any positive coefficients, and the initial values \(x_{-1},x_{-2},x_{-3}\) are any positive numbers. The author proves that every positive solution of the equation (\(E\)) is eventually periodic of (not necessarily prime) period \(T\), which is explicitly determined by the coefficients \(A,B\) and \(C\).