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I found this proof on Wiki

There I dont understand this part:
"Because $1 < 2^{r+1} < 2^{k+1}$ it follows that $2^{k+1}$ is not prime." How? Please explain

Carser
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tan
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    This question might help: http://math.stackexchange.com/questions/140804/if-2n1-is-prime-why-must-n-be-a-power-of-2 – iadvd Oct 05 '16 at 02:07

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In the previous line they showed that $2^r+1$ is a factor of $2^k+1$. If $1<2^r+1<2^k+1$ then $2^k+1$ would have a positive factor other than $1$ and itself, so it would not be prime.

David
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