Multiplicative semigroups related to the 3x+1 problem

Recently Lagarias introduced the Wild semigroup, which is intimately connected to the 3x+1 conjecture. Applegate and Lagarias proved a weakened form of the 3x+1 conjecture while simultaneously characterizing the Wild semigroup through the Wild Number Theorem. In this paper, we consider a generalization of the Wild semigroup which leads to the statement of a Weak qx+1 Conjecture for q any prime. We prove our conjecture for q=5 together with a result analogous to the Wild Number Theorem. Next, we look at two other classes of variations of the Wild semigroup and prove a general statement of the same type as the Wild Number Theorem.