Let $\mathcal{Z}(n,a,b)=\frac{\Phi_n(a,b)}{\gcd (\Phi_n(a,b),n)}$ by the $n$-th Zsigmondy number to base $(a,b)$, where $\Phi_n(a,b)$ is the $n$-th homogeneous cyclotomic polynomial. Zsigmondy proved that these are (with finite exceptions) greater than $1$. I am curious about exploring when these numbers are prime (even seemingly-trivial statements):
Possible Questions: Let $a>b\in\mathbb{N}_{>0}$ with $\gcd (a,b)=1$.
Is $\mathcal{Z}(n,a,b)$ composite for infinitely many $n$? (Probably Yes, see edit 2)
Is there at least one $n$ such that $\mathcal{Z}(n,a,b)$ is prime?
Fix $n>2$*, is $\mathcal{Z}(n,a,b)$ prime for some $a,b$?
Note that for $\mathcal{Z}(n,2,1)$, the first question is a strictly weaker question than the open problem: are there infinitely many composite Mersenne numbers $2^p-1$? ($\mathcal{Z}(p,2,1)=2^p-1$)
The second question is related to this unanswered question when $b=1$, $p\nmid a-1$. ($\mathcal{Z}(p,a,1)=\frac{a^p-1}{a-1}$)
*$\mathcal{Z}(1,a,b)=a-b$, and $\mathcal{Z}(2,a,b)=a+b$ (when $2\nmid a-b$), and so are prime for infinitely many $a,b$ by Dirichlet's Theorem on arithmetic progressions. This seems to get much harder for $n>2$, and is a special case of (a weaker) Bunyakowski's Conjecture.
All of these simply stated questions seem hard to make progress on. Is there any approaches one has to these questions? Any heuristics?
Edit: Note that for $n\neq 2$, $\mathcal{Z}(n,a,b)=\Phi_n(a,b)$ except when for the greatest prime factor $p$ of $n$, $n=p^kj$, where $j$ is the first integer such that $p|a^j-b^j$, in which case $\gcd (\Phi_n(a,b),n)=p$, and $ \mathcal{Z}(n,a,b)=\frac{\Phi_n(a,b)}{p}$.
Edit 2: For the first question: For $b=1$, we have that all primes $p\nmid a$ divide $\mathcal{Z}(n,a,1)$ for some $n$, as $a^n\equiv 1\mod p$ for some $n$ by elementary group theory. By similarly considering $a^n\equiv b^n\mod p$, we can show that if $p\nmid a$ and $p\nmid b$, we have that $p$ divides $\mathcal{Z}(n,a,b)$ for some $n$. But then we have:
$\mathcal{Z}(n,a,b)$ is prime for all but finitely many $n\Longrightarrow \exists N\in\mathbb{N}$ s.t $\forall p>N, p=\mathcal{Z}(n,a,b)$ for some $n$. The following conclusion can likely be made rigorous: but $\{\mathcal{Z}(n,a,b)\}_n$ is much more sparse than the primes, so this cannot be true.
