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
- [math]\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\}. }[/math]
The function [math]\displaystyle{ T : \mathbb{Z} \to \mathbb{Z} }[/math] as defined below is used in the "shortcut" definition of the Collatz conjecture:
- [math]\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} }[/math]
The Collatz conjecture asserts that for each positive integer [math]\displaystyle{ n }[/math], there is some iterate of [math]\displaystyle{ T }[/math] with itself which maps [math]\displaystyle{ n }[/math] to 1, that is, there is some integer [math]\displaystyle{ k }[/math] such that [math]\displaystyle{ T^{(k)}(n)=1 }[/math]. For example if [math]\displaystyle{ n=7 }[/math] then the values of [math]\displaystyle{ T^{(k)}(n) }[/math] for [math]\displaystyle{ k = 1, 2, 3,... }[/math] are 11, 17, 26, 13, 20, 10, 5, 8, 4, 2, 1 and [math]\displaystyle{ T^{(11)}(7)=1 }[/math].
The relation between the 3x + 1 semigroup and the Collatz conjecture is that the 3x + 1 semigroup is also generated by the set
- [math]\displaystyle{ \left\{ \dfrac{n}{T(n)} : n\gt 0 \right\}. }[/math]
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
- [math]\displaystyle{ \left\{\frac{1}{2}\right\}\cup \left\{\frac{3k+2}{2k+1}:k\geq 0\right\}, }[/math]
which is also generated by the set
- [math]\displaystyle{ \left\{\frac{T(n)}{n}: n\gt 0\right\}, }[/math]
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
References
- ↑ 1.0 1.1 Applegate, David; Lagarias, Jeffrey C. (2006). "The 3x + 1 semigroup". Journal of Number Theory 117 (1): 146–159. doi:10.1016/j.jnt.2005.06.010.
- ↑ 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". Princeton University. https://web.math.princeton.edu/~caraiani/papers/semigroups.pdf.
- ↑ J.C. Lagarias (2006). "Wild and Wooley numbers". American Mathematical Monthly 113 (2): 97–108. doi:10.2307/27641862. https://www.maa.org/sites/default/files/pdf/upload_library/22/Ford/lagarias97.pdf. Retrieved 18 March 2016.
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