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arXiv · 2606.24688

The number of Sylow subgroups and a generalization of Mersenne primes

Jorge Jiménez Urroz·Alexander Moretó
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Abstract

Fix an integer $m$ bigger than 2. We prove that if there exists a finite group with $mp+1$ Suylow $p$-subgroups, where $p$ is large enough, then $mp+1$ is prime. This improves on a theorem of M. Hall and is a partial answer to Brauer's Problem 26. Our proof uses techniques from analytic number theory, and it also raises new questions in that area.

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Jorge Jiménez Urroz, Alexander Moretó. 2026-06-23. The number of Sylow subgroups and a generalization of Mersenne primes. https://arxiv.org/abs/2606.24688

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