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 $ .

  • 2
    $\begingroup$ 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

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.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.