3x + 1 semigroup

In algebra, the 3x + 1 semigroup is a special subsemigroup of the multiplicative semigroup of all positive rational numbers.^[1] The elements of a generating set of this semigroup are related to the sequence of numbers involved in the still open Collatz conjecture or the "3x + 1 problem". The 3x + 1 semigroup has been used to prove a weaker form of the Collatz conjecture. In fact, it was in such context the concept of the 3x + 1 semigroup was introduced by H. Farkas in 2005.^[2] Various generalizations of the 3x + 1 semigroup have been constructed and their properties have been investigated.^[3]

Definition

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

${\displaystyle \{2\}\cup \left\{{\frac {2k+1}{3k+2}}:k\geq 0\right\}=\left\{2,{\frac {1}{2}},{\frac {3}{5}},{\frac {5}{8}},{\frac {7}{11}},\ldots \right\}.}$

The function ${\displaystyle T:\mathbb {Z} \to \mathbb {Z} }$ as defined below is used in the "shortcut" definition of the Collatz conjecture:

${\displaystyle T(n)={\begin{cases}{\frac {n}{2}}&{\text{if }}n{\text{ is even}}\\[4px]{\frac {3n+1}{2}}&{\text{if }}n{\text{ is odd}}\end{cases}}}$

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

The relation between the 3x + 1 semigroup and the Collatz conjecture is that the 3x + 1 semigroup is also generated by the set

${\displaystyle \left\{{\dfrac {n}{T(n)}}:n>0\right\}.}$

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:^[1]

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

${\displaystyle \left\{{\frac {1}{2}}\right\}\cup \left\{{\frac {3k+2}{2k+1}}:k\geq 0\right\},}$

which is also generated by the set

${\displaystyle \left\{{\frac {T(n)}{n}}:n>0\right\},}$

is called the wild semigroup. The integers in the wild semigroup consists of all integers m such that m ≢ 0 (mod 3).^[4]

See also

• Wild number