The 3x+1 Semigroup

The 3x+1 semigroup is the multiplicative semigroup generated by the rational numbers of form (2k+1)/(3k+2) for non-negative k, together with 2. This semigroup encodes backward iteration under the 3x+1 map, and the 3x+1 conjecture implies that it contains every positive integer. We prove this is the case, and show that this semigroup consists of all positive rational numbers a/b such that 3 does not divide b.Comment: 16 pages, latex; minor change

Multiplicative semigroups related to the 3x+1 problem

AbstractRecently 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