I am looking for a reference or proof for the following problem:

Problem: Let $r$ be prime, then $2r$ is a Sylow $p$-number if and only if $2r=1+p^{2^n}$. Thanks in advance.

I am looking for a reference or proof for the following problem: Problem: Let $r$ be prime, then $2r$ is a Sylow $p$-number if and only if $2r=1+p^{2^n}$. Thanks in advance. |
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Sylow Numbers of Finite Groups(1995) by JP Zhang: "a natural number $n$ is said to be a Sylow number for a finite group $G$ if $n$ is the number of Sylow $p$-subgroups of $G$ for some prime $p$." In this same article (see: sciencedirect.com/science/article/pii/S0021869385712355) Zhang includes your claim, with its proof sourced to a manuscript of his under preparation, calledSylow Numbers of Finite Groups, II. Impossible Values.If this latter paper has appeared in the 17 year interim, it has been under a different name. – Benjamin Dickman Oct 13 '12 at 17:29