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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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You need to expalin what a "Sylow $p$-number"is. I have not seen the term before, but I am guessing it is an integer which is the number of Sylow $p$-subgroups of some finite group. Even if this is correct, I supect that most readers will not be familiar with the term. –  Geoff Robinson Aug 29 '12 at 18:33
According to 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, called Sylow 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
See also the article whose abstract appears here: zentralblatt-math.org/portal/en/zmath/en/search/…. The article itself should be findable online, but it has not been translated into English. If you can read Chinese, great; if not, try to find someone to translate it. If this becomes a dire matter and you can't find a translator, email me (info on my user-page) and I'll give it a shot. –  Benjamin Dickman Oct 13 '12 at 17:50

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