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

preprint2009arXivOpen access

We show that the equation in the title (with $Φ_m$ 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.

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

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Some Diophantine equations from finite group theory: Φm(x)=2pn−1Φ_m(x)=2p^n-1Φm​(x)=2pn−1

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