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Explanation Let $\mathbb{P}=\{p_1,p_2,\ldots\}=\{2,3,5,\ldots\}$ be the set of prime numbers. Recall that by the Fundamental Theorem of Arithmetic that any positive integer $n$ can be written in the form $$n=p_1^{e_1}p_2^{e_2}\ldots p_m^{e_m}$$ where $m$ and the $e_j$'s are non-negative integers. Define the function $\mathrm{gpf} \colon \mathbb{Z}^+\rightarrow \mathbb{P}$ where $\mathbb{Z}^+$ denotes the positive integers. Let $p$ and $q$ be positive integers. Choose an initial positive integer $x_0$ and define a set recursively by $$x_{n+1}=gpf(px_n+q).$$ The "gpf-conjecture" is the statement that no matter which $p$ and which $q$ are chosen, the set $\{x_n \colon n=0,1,2,\ldots\}$ is always a finite set. |