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Prime numbers, diophantine equations, diophantine approximations, analytic or algebraic number theory, arithmetic geometry, Galois theory, transcendental number theory, continued fractions

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The sequence $a_{n+1}=$ the greatest prime factor of $(xa_n+y)$

Let $\operatorname{ GPF}(n)$ be 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} …
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