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I denote by $\operatorname{ GPF}(n)$ the greatest prime factor of $n$, eg. $\operatorname{ GPF}(17)=17$, $\operatorname{ GPF}(18)=3$.

Is there a way to prove that the sequence $a_{n+1}=\operatorname{ GPF}(qa_n+p)$, eventually enter a cycle for all initial values of positive integers $q,a_0,p>1$?

Which my simulations seem to indicate - although the sequence $a_{n+1}=\operatorname{ GPF}(a_n^2+1)$ with $a_0=2$ appears to run off into infinity.

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    Also posted to http://mathoverflow.net/questions/88282/the-sequence-a-n1-the-greatest-prime-factor-of-xa-ny/88304#88304 where a reference has been given to an affirmative answer to the first question in the case $q=1$.2012-02-13

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Since this question has been answered on Mathoverflow by Gerry Myerson, I'm going to post a quote of the answer here for sake of removing this post from the Unanswered Questions queue.

There is a paper on this problem, Mihai Caragiu, Recurrences based on the greatest prime factor function, JP J. Algebra Number Theory Appl. 19 (2010), no. 2, 155–163, MR2796479 (2012a:11010). The summary begins,

We introduce and discuss a generalized ultimate periodicity conjecture for prime sequences $\lbrace q_n\rbrace_{n\ge0}$ in which every term $q_n$ is recursively defined as the maximum element of the finite set $\lbrace P(A_jq_{n-1}+B_j)\mid j=1,\dots,k\rbrace$, where $P(x)$ represents the greatest prime factor of $x$, while $A_j$ and $B_j$ are fixed positive integers for $1\le j\le k$.

I haven't seen the paper, just the summary in Math Reviews.

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    @LaplacianFourier, ... isn't it community wiki already?2016-08-20