# Primitive root modulo $4q +1$

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 $ord_{p} ( 2) = 2q$ .

• That would make 2 a quadratic residue modulo $p$. But considerations modulo 8 rule that out. – Gerry Myerson Sep 26 '13 at 9:35