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 $3$

toone 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*

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.

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