Zsigmondy's Theorem Generalization

Question

Zsigmondy's Theorem states that if $a>b>0$ are coprime integers then for any integer $n\geq 1$ there is a prime $p$ that divides $a^n-b^n$ and does not divide $a^k-b^k$ for any positive integer $k<n$ with the exceptions $n=a-b=1$, $n=2$ and $a+b$ is a power of $2$, and $n=6, a=2, b=1$. This also works if we replace $a^n-b^n$ with $a^n+b^n$.

I was wondering if there are any generalizations of this in the following direction: let $u,v,a,b\in \mathbb{Z}$ such that $a>b>0$, $u,v\neq 0$ such that $\gcd(ua, vb)=1$. Is it the case that (with finitely many or trivial exceptions) for any $n\geq 1$ there is a prime $p$ such that $p$ divides $ua^n+vb^n$ but $p$ does not divide $ua^k+vb^k$ for any $k<n$?

Thank you!

Answer 1

Given an integer sequence $(s_n)_{n \in \mathbb{N}}$, a prime number $p$ is said to be a primitive divisor of the $n$th term $s_n$ if $p$ divides $s_n$, but $p$ does not divide any $s_m$, with $m < n$.

Thus Zsigmondy's Theorem determines all the terms of the sequence $(a^n - b^n)_{n \in \mathbb{N}}$ (and $(a^n + b^n)_{n \in \mathbb{N}}$) that do not have a primitive divisor. This has been generalized to Lucas and Lehmer sequences, see "Yu. Bilu, G. Hanrot and P.M. Voutier, Existence of Primitive Divisors of Lucas and Lehmer Numbers" (https://hal.inria.fr/inria-00072867/document).

Unfortunately, characterizing the terms without primitive divisors of more general linear recurrences, like $(ua^n + vb^n)_{n \in \mathbb{N}}$, seems currently out of reach.