Let $\Phi_n(t)$ be the $n$-th cyclotomic polynomial over the rationals. Stephens proved in 1971 that $\Phi_p(q)$ and $\Phi_q(p)$ are not always coprime for primes $p,q$ since one has
$$ gcd(\Phi_p(q),\Phi_q(p)) = 2pq+1 $$
$$ p=17, q=3313 $$
See the wiki page about the Feit-Thompson Conjecture.
Question: Are there more solutions to
$$ \gcd(m,n)=1 $$
$$ \gcd(\Phi_m(n),\Phi_n(m)) = 2mn+1\ ? $$
None found for small $m,n.$
UPDATE: I have a new solution: $m=464$, $n=21$ for which I do not know why $r=2mn+1$ is also prime !
After several weeks of computations there are no more new solutions with $$ m \leq 13400. $$