In algebra, the 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]
The 3x + 1 semigroup is the multiplicative semigroup of positive rational numbers generated by the set
\{2\}\cup\left\{
| 2k+1 | |
| 3k+2 |
:k\geq0\right\}=\left\{2,
| 1 | |
| 2 |
,
| 3 | |
| 5 |
,
| 5 | |
| 8 |
,
| 7 | |
| 11 |
,\ldots\right\}.
The function
T:Z\toZ
T(n)=\begin{cases}
| n | |
| 2 |
&ifniseven\\[4px]
| 3n+1 | |
| 2 |
&ifnisodd\end{cases}
The Collatz conjecture asserts that for each positive integer
n
T
n
k
T(k)(n)=1
n=7
T(k)(n)
k=1,2,3,...
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 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 in lowest terms having the property that b ≠ 0 (mod 3). In particular, S contains every positive integer.
The semigroup generated by the set
| \left\{ | 1 |
| 2 |
\right\}\cup\left\{
| 3k+2 | |
| 2k+1 |
:k\geq0\right\},
which is also generated by the set
| \left\{ | 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]