We show that the equation in the title (with Φm​ the mth cyclotomic
polynomial) has no integer solution with n≥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.Comment: 17 pages, slightly extended version is available as Max-Planck
preprint MPIM 2009-6