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Months ago, I asked In the $x + 1$ problem, does every positive integer $x$ eventually reach $1$? I think it's fairly easy to adapt the arguments given in the answers to that one to prove that $x - 1$ always reaches $1$ as well.

It's also well known that $3x - 1$, $5x - 1$ and $5x + 1$ all have readily found cycles that don't include $1$. I think I also found cycles for $7x + 1$ but I seem to have misplaced the relevant notebook.

My question today is: which other variants of $3x + 1$ have been studied and proven to always or not always reach $1$? And is there a paper or a book that gathers a lot of the available research on the variants?

Mr. Brooks
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  • In the $x-1$-problem: isn't $0$ reached by $x_1=1$? So all trajectories arrive at $x_N=0$ for some $N$ and so enters a somehow "supertrivial" cycle... – Gottfried Helms Jul 09 '18 at 06:01

1 Answers1

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Write $$ b = { m \cdot a + 1 \over 2^A} $$ for one transformation with odd multiplicator $m$ and odd $a \to b$.
Then there is for all $m=2^M-1$ the trivial cycle $1 \to 1$ and for all $m=2^M+1$ the trivial cycle $-1 \to -1$
Besides the cycles you refer to, in the literature it is also known the cycle with $m=181$ on $a=27$, $b=611$ (I think) and I found a second one on $a=35$.
I didn't find any more cycle - either numerically with tests up to $m$ some thousands and projected cycle lengthes up to some 100. Also I didn't find something more in literature. (Btw. shouldn't this all be in a section in wikipedia's Collatz-article under "generalization"? Curious - I'll see later, I'm just in a holiday)
Note, that allowing negative $m$ we find two more $m$ allowing small cycles, but have it not at hand, see some of my recent questions/answers concerning the collatz-problem.

Update Inspired by the finding in an arxiv-article linked by some recent Q&A about cycles in a $7x \pm 1$ - problem, defined by $$ f(n) = \left \lbrace \begin{matrix} n/2 & \text{if $n$ is even} \\ 7n +1& \text{if } n \equiv 1 \pmod 4) \\ 7n -1& \text{if } n \equiv 3 \pmod 4) \\ \end{matrix}\right.$$
$ \qquad $which can also be rewritten as $ b = { 7 \cdot a + (2 - a \% 4) \over 2^A} $ for one transformation where the $\%$-sign denotes the residue function with modulo $4$ (often called mod in programming languages) I looked at the obvious generalizations with $m=\{3,5,7,9,11,13,15,17,19\}$ (of course with the meaningful adaption of the $a \% 4$-rule) and found the following cycles testing small numbers: 
 m      cycles, (?likely) divergences    
 ----+------------------------------------------
 3      1,1,...    
 5      1,1,...   
           7,9,11,7,...   
 7      1,1,...       
 9      1,1,...     
           13, 29, 65, 73, 41, 23, 13, ...  
           (? divergences...)
11      1,3,1,... 
           (?divergences)
13      1,3,5,1,...    
           25, 81, 263, 855, 2779, 1129, 3669, 2981, 1211, 123, 25, ...
           49, 159, 517, 105, 341, 277, 225, 731, 297, 965, 49 ,...    
           (?divergences)
15      1,1,... 
           (?divergences)           
17      1,1,...   
           (?divergences)           
19         (?divergences)      
181      27,611,27,... 
           35,99,35, ...
           (?divergences)

All found cycles have exact counterparts in the negative numbers.

Gottfried Helms
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