I am cross-posting this from math.stackexchange, as it has received upvotes but no comments/answers after a couple months.
Let $\mathcal{Z}(n,a,b)=\frac{\Phi_n(a,b)}{\gcd (\Phi_n(a,b),n)}$ be 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 bottom of post)
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 $a,b$?
Are there any approaches to these or tangential questions in the literature? Does anyone have other ideas? Heuristics?
Possible motivations:
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.
More notes: 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}$.
Also 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.
Edit: A paper by Schinzel and Davenport provides an alternative (and partially constructive) proof of the first question on $\mathcal{Z}(n,a,b)$ being composite for a fixed $(a,b)$, utilizing a sort of Aurifeuillian factorization for certain $n$. Note that $\mathcal{Z}(n,a,b)$ is precisely the product of primitive prime divisors of $a^n-b^n$.
