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We show that the equation in the title (with $\Phi_m$ the $m$th cyclotomicpolynomial) 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 theoreticalinvestigation by Z. Akhlaghi, M. Khatami and B. Khosravi.

Some Diophantine equations from finite group theory: $\Phi_m(x)=2p^n-1$