*Remark : I've found a rather trivial answer for this question and so very likely the premise of paralleling it with the Zsigmondy-theorem is wrong, so this question might better be retracted. I'll give it a certain time loking at reactions on the [discussion in the MO-meta site](https://meta.mathoverflow.net/questions/3383/retract-a-question-when-its-resolution-comes-out-to-be-trivial-and-some-premise).  

Following a hint in META I added a comment where you can vote for deletion.*




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I'm (recreationally) studying properties of the generalized Collatz-problem ("gCp" for shortness) for $3x+\rho$ and look for help in the question of the existence of cycles in gCp's with composite $\rho$ as parameter in relation to that in gCp's with the primefactors of $\rho$ as parameters. 
To explain the problem formally, I'd like to use here the compact "Syracuse"-notation of the gCp in the following way:
$$ T\_\rho(a;[A]) := {3a+\rho \over 2^A} \qquad \qquad \text{where } A=v\_2(3a+\rho)
$$
with odd positive integer in the parameter $\rho$. By that definition the transformation $b=T(a;[A])$ works on odd numbers $a,b$ only.  

For an iterated transformation of $N$ steps I write simply $a\_{N+1}=T\_\rho(a\_1;[A\_1,A\_2,...,A\_N]) $. I denote $S$ as the sum of exponents $A\_k$ and to make the following formulae short, I define $E\_{N,S} = [A\_1,A\_2,...,A\_N]$ as the vector for a fixed set of exponents $A\_k$ having length $N$ and sum $S$.


I'm interested in the question on existence of cycles in $T\_\rho()$ for some composite $\rho$ in relation to that in $T\_\varphi()$ where $\varphi$ is a primefactor of $\rho$ .


For example let $\varphi, \sigma \in \Bbb P$ being prime and $\rho=\varphi \cdot \sigma$ being composite. Let some vector $E\_1 = E\_{N\_1,S\_1}$ be such that $a\_{N+1} = T\_\varphi(a\_1;E\_1)=a\_1$ is a cycle.  

It is easy to derive that then also $ b\_1 = T\_\rho(b\_1;E\_1)$ with $b\_1= \rho/\varphi \cdot a\_1$ is a cycle in $T\_\rho()$.  

Of course the analogue is true for a cycle in $T\_\sigma()$ in relation to $T\_\rho()$ with another set of exponents $E\_2 := E\_{N\_2,S\_2}$


* a) So we have first -by some simple analysis-, that a generalized Collatz-problem gCp with $T\_\rho()$ with a composite parameter $\rho$ defines the same cycles as the gCp's with the primefactors of $\rho$ as parameters (where only its elements $a\_k$ are rescaled by some factor as said above).


The two interesting parts are the following which I'd like to understand/to prove.


* b) The gCp of a composite parameter $\rho$ seems to have *in general* always additional cycles besides that of the gCp's defined by $\rho$'s primefactors.  

This reminds me of the [theorem of Zsigmondy](https://en.wikipedia.org/wiki/Zsigmondy%27s_theorem) about the primefactorization of Mersenne-numbers (where he proves the existence of the then so called "primitive primefactors"). Possibly it can be proven the same way, but I could not follow his proof. Let's call that additional cycles analoguously "primitive cycles".
* c) Different from the "general" case: when we have the primefactor $\sigma=3$ involved, the composition $\rho=\varphi \cdot 3$ with that primefactor $T\_{\varphi \cdot 3}()$ seems to *not* to lead to such additional "primitive cycles" - again in analogy to Zsigmondy's observation on the Mersenne-number $M\_6 = M\_{2 \cdot 3}$ which has no "primitive" factors.  

Of course I do such a statement only based on heuristics and the basic ***assumption that the Collatz-conjecture for $T\_1()$ is true*** and that it has only the "trivial cycle" $1=T\_1(1;[2,2,2,...,2])$.



> 
> So my question asks for help in proving the observations in b) and c) (under assumption of truth of the Collatz-conjecture). Perhaps it would be sufficient to only get Zsigmondy's derivations transparent enough.
> 
> 
> 




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\*Remark: for some examples and possibly for a better explanation of the elementary analysis of this you might look at the short treatize [at my webspace][3] (instead of the term "primitive cycles" I've used "unexplained cycles" there)\*

A Zsigmondy-theorem-analogy in the generalized Collatz-problem $3x+\rho$?
*The answers for b) and c) came out to be trivial and have likely nothing to to with Zsigmondy, so possibly I **should retract** my question.* 




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For the definition of a cycle for some $a\_1=T\_\rho(a\_1;E\_{N,S})$ we have the formula (6.b) in the referred article:
$$ a\_1 = \rho \cdot { Q(E\_{N,S}) \over 2^S - 3^N} \tag 1
$$
Let $\frac uv$ with $v>1$ denote the fully cancelled fraction in (1) then by
$$ a\_1 = \rho \cdot \frac uv \tag 2
$$
it is obvious that we can have a cycle at $a\_1$ only if $v | \rho$. Now let $v$ be prime, then setting $\rho=v$ makes this having an integral solution and thus having a cycle at $a\_1$. But of course *any* composite $\rho = v\cdot t$ makes this having an integral solution (and having a cycle at $t \cdot a\_1$) as well. So a gCp with composite $\rho$ has cycles at the same exponentsvector $E\_{N,S}$ as all gCp's with parameters being primefactor of $\rho$. 


At question b): the "unexplained" (or "primitive") cycles at some $E\_{N,S}$ occur if in (2) the denominator is composite and not prime - because in the gCp's with prime parameter the denominator $v$ is not completely cancelled, and thus that gCp's cannot have a cycle at that same $E\_{N,S}$  

This explains question b). 


The question c) is simply explained by the additional fact, that the denominator in (2) cannot have the primefactor $3$ - so for any composite $\rho $ the possession of primefactor $3$ does not introduce any new ("unexplained","primitive") cycle. 


So this seems to have nothing to do with Zsigmondy's theorem on mersenne-numbers. 


