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Let $q$ be a prime such that $ p = 1 + 4q $ is a prime as well. Show that $2$ is a primitive root modulo $p$ ( i.e. that $2$ generates the multiplicative group $( \mathbb{Z} / p\mathbb{Z})^{*}$ ) .

I don't understand how to exclude the case $ord_{p} ( 2) = 2q $ .

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    $\begingroup$ That would make 2 a quadratic residue modulo $p$. But considerations modulo 8 rule that out. $\endgroup$ – Gerry Myerson Sep 26 '13 at 9:35
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You can read the intro of Murty's paper http://www.math.ucsb.edu/~agboola/teaching/2005/winter/old-115A/murty.pdf, in particular it's explained in the second half of page 4.

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