Beyond Collatz: A $5n+1$ conjecture? [closed]

Question

Let

$$x_{n+1} = \begin{cases} x_n/2 &;\text{if } x_n \equiv 0 \pmod{2}\\ k\,x_n+1 &; \text{if } x_n\equiv 1 \pmod{2} \end{cases}$$

and $k=3$ and $x_n\in\Bbb N$. Collatz conjectured for this recurrence system that starting with any $x_n^{(start)}$ the system converges to a limit cycle (an atractor, orbit) of period $3$: $$\dots\rightarrow16\rightarrow8\rightarrow4\rightarrow2\rightarrow1\to4\to2\to1$$

Independent of the above allow me conjecture the following:

Let

$$x_{n+1} = \begin{cases} x_n/2 &;\text{if } x_n \equiv 0 \pmod{2}\\ k\,x_n+1 &; \text{if } x_n\equiv 1 \pmod{2} \end{cases}$$

and $k=5$ and $x_n\in\Bbb N$.

(Part 1) For this recurrence system starting with any $x_n^{(start)}$ the system either

• converges (stable) to the following limit cycle (an attractor of a repeating sequence, orbit) of period $7$: $$\dots\to16\to8\to4\to2\to1\to 6\to3\to16\to8\to4\to2\to1$$ examples for this are when we start the recurrence with one of the following integers $3$, $15$, $19$, $51$, $65$, $97$, $137$, $155$, $163$, $175$
• or converges (stable) to a limit cycle (an attractor of a repeating sequence, orbit) of period $10$. Example: $$\dots\to13\to 66\to 33\to 166 \to 83 \to 416\to 208\to 104\to 52\to26\to 13\to \dots$$ examples for this are when we start the recurrence with one of the following integers $5$, $13$, $17$, $27$, $33$, $43$, $83$
• or diverges (intsable, ecape to infinity)

(Part 2) Hence if the Collatz conjecture would be true then the transition from $k=3$ to $k=5$ would represent a bifurcation from one single limit cycle of period $3$ to one specific limit cycle of periods $7$ (see above) and some other limit cycles of each period $10$.

Question (1): Could you contradict the above conjecture with a counter example?

Question (2): Is this conjecture genuine or has it been stated exactly like this in literature earlier?

citation Vaseghi 2013

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At reqeust for a Heuristic below a Mathematica program that we applied at TrueNorth Research.

ClearAll[collatz];

collatz1 = 1;

collatz[n_ /; EvenQ[n]] := (Sow[n];collatz[n/2])

collatz[n_ /; OddQ[n]] := (Sow[n]; collatz[5 n + 1])

runcoll[n_] := Last@Last@Reap[collatz[n]]

runcoll[13]

you can change $13$ with any other integer.

Answer 1

The reason why there should be values that escape to infinity in this case is that if one considers a so to say "random model" for the $k=3$ variant one has roughly speaking a change of the size by roughly $3/2$ half the time and a change of $1/2$ half-the time, and the product being less than $1$ one would expect a long-term decay.

By contrast, for the current variant, one has an increase by about $5/2$ half the time and a decrease of $1/2$ half the time. The product being larger than one one would expect a long term increase.

Thus, one would expect there are some values where the function will escape to infinity (so option three should occur sometimes).

Also it seems there is $17 \to 86 \to 43 \to 216 \to 108 \to 54 \to 27 \to 136 \to 68 \to 34 \to 17$.

Added: again I misunderstood the question. As it leaves opent the existence of other such cycles, and only fixes their lengths. The one in OP and the one I recall are it seems known to be the only ones of this length.

It seems feasible there are actually no others if one does not find any somewhat soon. The existence of certain cycles is excluded in a paper mentioned below.

To consider this variant is not original, for example it is mentioned in passing in a blog post by Tao The Collatz conjecture, Littlewood-Offord theory, and powers of 2 and 3 with a more sophisticated form of the argument above (for escaping values).

Also an older math.SE questions (I think not the one mentioned in comments; added: I meant the one by OP, the one by David Speyer mentioning it appeared while I edited) discusses this precise problem giving some additional information and references https://math.stackexchange.com/questions/14569/the-5n1-problem specfically to some paper of Metzger that determines which scyles of certain lengths can exist (also mentioning the one reproduced above, in addition to the one in OP).