($\textbf{gpf}$ here is the function returning the greatest prime factor of an integer.)
Consider the family of recurrence relations given by
$$s_{i+1} := \text{gpf}(a s_i + 1).$$
I don't have proof of this (is there any?), but it seems extremely likely from experimentation that such sequences will fall into a cycle for any positive integer $a>1$ and any starting value $s_1$.
e.g. $s_1 = 2, a=5$ gives:
$ \text{gpf}(5(2)+1) = 11 \\ \text{gpf}(5(11)+1) = 7 \\ \text{gpf}(5(7)+1) = 3 \\ \text{gpf}(5(3)+1) = 2 \\ \text{gpf}(5(2)+1) = 11 \\ \ldots $
Does this result follow in general for analogous quadratic sequences? Specifically,
$$s_{i+1} :=\text{gpf}(s_i^2 + k)$$
for positive integers $k$ and initial value $s_1$.
It is definite that some cycles do exist:
$$ \begin{array}{|ll|ll|} \hline \textbf{k} & & \textbf{cycle} & \\ \hline 1 && 89,233 & \\ \hline \text{(also 1)} && 19121,10753313,1241761,3817193,107837 & \\ \hline 2 && 571,2131 & \\ \hline 3 && 2089,10009 & \\ \hline 4 && 5741,33461 & \\ \hline 5 && 3,7 & \\ \hline 6 && & \\ \hline 7 && 2,11 & \\ \hline 8 && 19,41,563,83 & \\ \hline 9 && 6197,7109,6373 & \\ \hline 10 && 41,89,103 & \\ \hline 11 && 3,5 & \\ \hline 12 && 181,2521 & \\ \hline 13 && & \\ \hline \end{array} $$
I've omitted $6$ and $13$ as I've been unable to find a cycle yet for them; if a cycle does exist for $k=6$, its minimum value is almost certainly larger than $10^7$.
So, I am asking:
- If there's a known result along these lines.
- If not, any compelling argument for or against there being such cycles for all $k$.
I am also interested in whether every $s_1$ leads to a cycle for a given $k$, assuming a cycle is known to exist. For example, it seems plausible to me that many large values of $s_1$ for $k=1$ grow fast enough that they may 'escape' rather than always be pulled down to the $\{89,233\}$ cycle.
I recognize this question seems at least distantly related to Collatz, and thus an answer may be flatly unknowable at this time. If that's the case, so be it.