# Questions tagged [primitive-roots]

For questions about primitive roots in modular arithmetic, index calculus, and applications in cryptography. For questions about primitive roots of unity, use the (roots-of-unity) tag instead.

394 questions
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### n is primitive root for odd prime p. If -n is primitive root iff p = 4k+1 for some integer k

if $n$ is primitive root for odd prime $p$, then $-n$ is also a primitive root for $p$ $\iff p = 1 \text{ mod } 4$ I am having trouble solving this question, can someone show to me how to solve it? ...
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### $\zeta$ is a p-th primitive untary root iff $-\zeta$ if a 2p-th primitive unitary root, with $p$ an odd prime

$\Rightarrow$ $(-\zeta)^{2p}=(-\zeta^p)^{2}=1$ and if $i \in[{1,2p-1}]$ there is $k \in[0,p-1]$ such that $i=2k+1$ if is odd and $2k$ if is even so $(-\zeta)^{i}=(-\zeta)^{2k}=(\zeta^k)^{2}\neq 1$ ...
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### Finding primitive roots modulo $x$

I'm just starting with number theory and by now I know how to test whether a given number $\alpha$ is a primitive root mod $p$ or not. But I'm not sure yet how it works if $\alpha$ isn't given. I saw ...