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

TL;DR

This paper proves the non-existence of integer solutions for specific Diophantine equations involving cyclotomic polynomials, which are relevant in finite group theory, thereby advancing understanding in this mathematical area.

Contribution

It establishes new non-existence results for particular Diophantine equations derived from finite group theory, specifically those involving cyclotomic polynomials and prime powers.

Findings

No integer solutions for the specified equations with given parameters.

The results connect Diophantine equations to finite group theoretical problems.

Provides methods potentially applicable to other similar equations.

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.