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.
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.
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.
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 $2$.)