MHB Primitive Roots Modulo $p$: The $(p-1)/2$ Rule

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The discussion centers on the condition for a number \( g \) to be a primitive root modulo an odd prime \( p \). It is clarified that \( g \) is a primitive root modulo \( p \) if and only if its order is \( p-1 \). The statement that \( g^{(p-1)/2} \equiv -1 \pmod{p} \) does not guarantee \( g \) is a primitive root, as demonstrated by the example of \( g = 6 \) for \( p = 7 \), which satisfies the congruence but is not a primitive root. The conclusion emphasizes that while the congruence indicates the order is not \((p-1)/2\), it does not rule out other possible orders. Thus, the original assertion is incorrect.
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Is it true that $g$ is a primitive root modulo $p$ if and only if $g^{(p-1)/2} \equiv -1 \pmod p$?
 
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Alexmahone said:
Is it true that $g$ is a primitive root modulo $p$ if and only if $g^{(p-1)/2} \equiv -1 \pmod p$?
No. For example, take $p=7$ and $g=6$. The congruence is satisfied, but $6$ is not a primitive root$\mod 7$.
 
Alexmahone said:
Is it true that $g$ is a primitive root modulo $p$ if and only if $g^{(p-1)/2} \equiv -1 \pmod p$?

Hi Alexmahone,

I assume that $p$ is supposed to be an odd prime?

If so, then it is true that $g$ is a primitive root modulo $p$ if and only if the order of $g$ is $p-1$ modulo $p$.

So what we need is that the order of $g$ is $p-1$.
From $g^{(p-1)/2} \equiv -1 \pmod p$, we can only conclude that the order of $g$ is not $(p-1)/2$, but it could still be $(p-1)/3$ or some such.

Opalg's example is showing exactly that, which is the simplest counter example. He picked the smallest odd prime for which $(p-1)/k$ is an integer with $k>2$, and he found a $g$ to match. :)
 
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