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

Abstract

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.

arXiv comment17 pages, slightly extended version is available as Max-Planck preprint MPIM 2009-62