# Repeating cycles in the $3n-1$ problem

While tracking sequences beginning with 1-to-3 digit integers, I have found 3 different repeating cycles in the $3n-1$ problem (similar to the Collatz Conjecture). They are 1, 2, 1..., 5, 14, 7, 20, 10, 5..., and 17, 50, 25, 74, 37, 110, 55, 164, 82, 41, 122, 61, 182, 91, 272, 136, 68, 34, 17....

Has there been much research into this problem? If so, has anyone found any other such sequences? I have checked starting values up to approximately 150.

• If your sequence terminates, then you can't have the last two repeating sequences of leading digits you mentioned going on forever. Are your sequences actually forming cycles out of the true values? Or do they terminate, and you're just pointing out that you get a certain cycle of leading digits "until almost the end"? Or did your computer run out of time before you could figure out termination? Commented Apr 23, 2014 at 18:34
• No, it does not terminate, it only stabilizes to a cycle of repeating digits. It is easy to show that they always continue cycling for as long as one cares to generate the sequences. Also, I generated those sequences by hand; I do not use a computer for this. Commented Apr 23, 2014 at 19:42
• I first misunderstood your problem and deleted my comment. What are you proposing to be the problem? More precisely: What statement do you want to prove or disprove? or otherwise: What heuristic do you want to have? Commented Apr 23, 2014 at 19:44
• Your cycles have a very close link with the cycles of $3n+1$ starting with negative $n$. Commented Apr 23, 2014 at 22:36
• Commented Apr 24, 2014 at 1:20

## 2 Answers

To spell out in more detail what I hinted at in a comment:

Consider the standard Collatz mapping $x \rightarrow 3x + 1$ applied to negative values. We get $-x \rightarrow -3x + 1$. So the standard Collatz iteration over negative values is isomorphic to iterating $x \rightarrow 3x - 1$ over positive values.

The cycles you've identified are the only known ones, and while I don't know the state of the literature I assume that if it had been proven that there were no more negative Collatz cycles then that fact would be mentioned in the Wikipedia article.

I don't remember longer articles about this specific variant; but if there is some notable article (before 2010, I think), then you should find it in J. Lagarias' "annotated bibliography" which I could download in various updates online over the years. I recommend the Lagarias byibliography as an entrance point.

A short overview, testing starting values initially $$a_1 \le 1000$$ and then later in an update by the construction-principles of the cycles and moreover for the generalization to $$mx \pm1$$ for $$m \lt 1.14E20$$ shows a very sparse occurence of cycles in any generalization of the Collatz-problem, and confirms your finding.
For the overview see a more precise description at the header of the picture: