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


$$
gcd(\Phi\_p(q),\Phi\_q(p)) = 2pq+1
$$


for


$$
p=17, q=3313
$$


See the wiki page about the [Feit-Thompson Conjecture](https://en.wikipedia.org/wiki/Feit%E2%80%93Thompson_conjecture).


Question: Are there more solutions to


$$
\gcd(m,n)=1
$$


and


$$
\gcd(\Phi\_m(n),\Phi\_n(m)) = 2mn+1\ ?
$$


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
$$
m \leq 13400.
$$



Variant of Stephens result $\gcd(\Phi_p(q),\Phi_q(p))=2pq+1$ for $p=17$, $q=3313$
You may want to look at Karl Dilcher and Joshua Knauer, On a conjecture of Feit and Thompson, in the book, High Primes and Misdemeanours: Lectures in Honour of the 60th Birthday of Hugh Cowie Williams, 169–178, Fields Inst. Commun., 41, Amer. Math. Soc., Providence, RI, 2004, MR2076245 (2005c:11003). In addition to a thorough discussion of the state of the art as regards the Feit-Thompson conjecture, they discuss the case you appear to be interested in, where the variables are not necessarily prime. They find, for example, that if $p$ is an odd prime, then under various conditions on $k$ we get $\gcd(\Phi\_2(kp),\Phi\_{kp}(2))=kp+1$. 


