  We show that the equation in the title (with $\Phi_m$ the $m$th cyclotomic
polynomial) has no integer solution with $n\ge 1$ in the cases $(m,p)=(15,41),
(15,5581),(10,271)$. These equations arise in a recent group theoretical
investigation by Z. Akhlaghi, M. Khatami and B. Khosravi.
