Variant of Stephens result $\gcd(\Phi_p(q),\Phi_q(p))=2pq+1$ for $p=17$, $q=3313$ 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.
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
$$
m \leq 13400.
$$
 A: 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  $\gcd(\Phi_2(kp),\Phi_{kp}(2))=kp+1$. 
A: Probably none others with $\min(p,q)<10000$ and $\max(p,q)<20000$. For $p,q \lt 4000$ there are $59$ cases of $\Phi_p(q) \mod 2pq+1=0.$ In all those cases $2pq+1$ is prime. Taking that as an additional requirement, there are $201$ cases with the bounds as above but no cases in which both congruences hold. Note that $\Phi_q(p)=\frac{p^q-1}{p-1}$ so to check if this is a multiple of $m=2pq+1$ it is enough to check if $p^q \mod m=1$ and that is a quick calculation.
