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Let $k \geq 2$ be a positive integer and let $n=2^k+1$. How can I prove that $n$ is a prime number if and only if $3^{\frac{n-1}{2}} \equiv -1 \pmod n.$

Fixed.

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    The easier half of this proposition is established for [this newer Question](http://math.stackexchange.com/questions/1815066/prove-that-3q-1-2-equiv-1-pmod-q-then-q-is-prime-number). Since neither implication is really detailed here, I don't see treating either of these Questions as a duplicate.2016-06-06

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Here are two options for finding a proper proof for this theorem (called Pepin's test).

1) http://en.wikipedia.org/wiki/Pepin's_test.

2) "Solved and Unsolved Problems in Number Theory" by Daniel Shanks. This book includes the proof for that theorem.

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This is the simplest case of Pratt certificates for primality - have a look at http://mathworld.wolfram.com/PrattCertificate.html for a better explanation. (In the notation of the article, your question corresponds to the case where the only $p_i$ is $2$.)