On the length of largest cycle of quadratic dynamical systems modulo a prime

Sammanfattning: In this paper, we investigate dynamical systems which are given by f : x 7→ x2 + c modulo a prime and for which value of the constant c can we get the largest possible cycle. We get the main ideas for finding cycles in iteration functions by introducing Floyd’s algorithm. Next, we implement the algorithms and ideas for finding cycles in Mathematica and visualize the results. In addition, we study the theoretical bounds for the length of the largest cycle and the case of c = 0 in detail. At last, we have some short discussion to this effect.