3x 1 semigroup

Definition

The 3x + 1 semigroup is the multiplicative semigroup of positive rational numbers generated by the set

The function T : Z → Z, where Z is the set of all integers, as defined below is used in the Collatz conjecture:

The Collatz conjecture asserts that for each positive integer n, there is some iterate of T with itself which maps n to 1, that is, there is some integer k such that T^(k)(n) = 1. For example if n = 7 then the values of T^(k)(n) for k = 1, 2, 3, . . . are 22, 11, 34, 17, 52, 26, 13, 40, 20, 10, 5, 16, 8, 4, 2, 1 and T⁽¹⁶⁾(7) = 1.

The relation between the 3x + 1 semigroup and the Collatz conjecture is that the 3x + 1 semigroup is also generated by the set { n T ( n ) : n > 0 } .

The weak Collatz conjecture

The weak Collatz conjecture asserts the following: "The 3x + 1 semigroup contains every positive integer." This was formulated by Farkas and it has been proved to be true as a consequence of the following property of the 3x + 1 semigroup. "The 3x + 1 semigroup S equals the set of all positive rationals a/b in lowest terms having the property that b ≠ 0 (mod 3). In particular, S contains every positive integer."

The wild semigroup

The semigroup generated by the set { 1 2 } ∪ { 3 k + 2 2 k + 1 : k ≥ 0 } , which is also generated by the set { T ( n ) n : n > 0 } , is called the wild semigroup. The integers in the wild semigroup consists of all integers m such that m ≠ 0 (mod 3).