The 3x + 1 semigroup is the multiplicative semigroup of positive rational numbers generated by the set
The function as defined below is used in the "shortcut" definition of the Collatz conjecture:
The Collatz conjecture asserts that for each positive integer , there is some iterate of with itself which maps to 1, that is, there is some integer such that . For example if then the values of for are 11, 17, 26, 13, 20, 10, 5, 8, 4, 2, 1 and .
The relation between the 3x + 1 semigroup and the Collatz conjecture is that the 3x + 1 semigroup is also generated by the set