On the period five trichotomy of all positive solutions of xn+1 = (p +xn−2)/xn

We study the behavior of all positive solutions of the difference equation in the title, where p is a positive real parameter and the initial conditions x−2,x−1,x0 are positive real numbers. For all the values of the positive parameter p there exists a unique positive equilibrium ¯x which satisfies the equation ¯x2 = ¯x +p. We show that if 0 < p < 1 or p 2 every positive bounded solution of the equation in the title converges to the positive equilibrium ¯x. When 0 < p <1 we show the existence of unbounded solutions. When p 2 we show that the positive equilibrium is globally asymptotically stable. Finally we conjecture that when 1