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 yet to be proved conjecture known as the 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
The function T : Z → Z, where Z is the set of all integers, as defined below is used in the "shortcut" definition of 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 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 .
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 , which is also generated by the set , 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
- ^ ^{a} ^{b} 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. 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. doi:10.2307/27641862. Retrieved 18 March 2016.