$3 x + 1$ semigroup

In algebra, the $3 x + 1$ 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

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

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 $n$ , 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 11, 17, 26, 13, 20, 10, 5, 8, 4, 2, 1 and $T^{(11)}(7)=1$ .

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

\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 ⁠ab⁠ 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

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

which is also generated by the set

\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]

References

^ ^a ^b Applegate, David; Lagarias, Jeffrey C. (2006). "The $3 x + 1$ semigroup". Journal of Number Theory. 117 (1): 146–159. doi:10.1016/j.jnt.2005.06.010. MR 2204740.
^ H. Farkas (2005). "Variants of the 3 N + 1 problem and multiplicative semigroups", Geometry, Spectral Theory, Groups and Dynamics: Proceedings in Memor y of Robert Brooks. Springer.
^ Ana Caraiani. "Multiplicative Semigroups Related to the 3x+1 Problem" (PDF). Princeton University. Retrieved 17 March 2016.
^ J.C. Lagarias (2006). "Wild and Wooley numbers" (PDF). American Mathematical Monthly. 113 (2): 97–108. doi:10.2307/27641862. JSTOR 27641862. Retrieved 18 March 2016.

[Applegate-1] Applegate, David; Lagarias, Jeffrey C. (2006). "The $3 x + 1$ semigroup". Journal of Number Theory. 117 (1): 146–159. doi:10.1016/j.jnt.2005.06.010. MR 2204740.

[2] H. Farkas (2005). "Variants of the 3 N + 1 problem and multiplicative semigroups", Geometry, Spectral Theory, Groups and Dynamics: Proceedings in Memor y of Robert Brooks. Springer.

[Ana-3] Ana Caraiani. "Multiplicative Semigroups Related to the 3x+1 Problem" (PDF). Princeton University. Retrieved 17 March 2016.

[4] J.C. Lagarias (2006). "Wild and Wooley numbers" (PDF). American Mathematical Monthly. 113 (2): 97–108. doi:10.2307/27641862. JSTOR 27641862. Retrieved 18 March 2016.

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[2]

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Definition

The weak Collatz conjecture

The wild semigroup

See also

References