id like to start answering this question y forst addressing the question in my answer so taht the reader knows what I am seeking to do : Using the dynamical system and by considering the number of distinct p-cycles for when p is prime, give dynamical reasoning to prove the number theoretic result that the number 2 p1 1, for p 3 and p prime, has p as a factor (ie, p | (2p1 1))