The Mersenne prime $M_p=2^p-1$ can be expressed as $M_p=\Phi_p(2)$ where $\Phi_m(x)$ is a cyclotomic polynomial. It seems useful to generalize the Mersenne prime search to more general cyclotomic primes. Let $\Phi_m(x,y)=\Phi_m(x/y)y^{\phi(m)}$ be the homogenized cyclotomic polynomial. Since $\Phi_m(x)$ are irreducible, there should be infinitely many primes of the form $\Phi_m(r,s)$ as $r,s$ varies, by Schinzel's conjecture. One expect this should still be the case if we fixed $r,s$ with $|s| < r$ and $gcd(r,s)=1$ and varies $m$ instead. For example, are there infinitely many primes of the form $\Phi_m(2,1)$ ? This is a larger set than the Mersenne primes so is probably easier.
We searched recently for primes of the form $\Phi_{p_1..p_k}(r,s)$ over distinct odd primes $p_1,...,p_k$ and found a $2152$ digit prime $\Phi_{2021}(4,13)$ (which justifies this post). Cyclotomic numbers like Mersenne numbers have a simple product form $\Phi_m(r,s)=\prod_{d|m} (r^d-s^d)^{\mu(n/d)}$ which follows from $x^m-1=\prod_{d|m} \Phi_d(x)$ by inclusion/exclusion, so we can express, in nicer form $$\Phi_{2021}(4,13)=\frac{(4-13)(4^{2021}-13^{2021})}{(4^{43}-13^{43})(4^{47}-13^{47})}.$$ We also found a three-primes $5599$ digit example $$\Phi_{13.17.29}(11,-4)=\frac{(11^{6409}+4^{6409})(11^{13}+4^{13})(11^{17}+4^{17})(11^{29}+4^{29})}{(11+4)(11^{221}+4^{221})(11^{377}+4^{377})(11^{493}+4^{493})}.$$ which may seems surprising because the RHS must cancel out and be left with a single term. $\Phi_{3.7.11.17}(2,1)$ is also prime. Are there known examples with $k > 4$?
The formula for example for $k=2$, $\Phi_{p.q}(r,s)=\frac{(r^{p.q}-s^{p.q)}(r-s)}{(r^p-s^p)(r^q-s^q)}$ means we are searching along prime exponents which does not seem to be governed by the usual conjectures.
It does not seem easy to prove that there is at least one such prime for fixed $r,s$ with $gcd(r,s)=1$. Maybe the only way is to prove positive density but they seem very sparse.
