I'm finding all the primitive roots of $3^3 =27$. I know that if $r$ is a primitive root of $p$, then it is also primitive root of $p^k$. But how do we find all candidates of primitive roots of the form $p^k$ ?
Asked
Active
Viewed 510 times
0
-
See http://math.stackexchange.com/questions/1104189/primitive-roots-ga-modp – lab bhattacharjee Apr 06 '15 at 17:07
-
The reason I suggest the above as a duplicate is that the answer is "it's hard even for primes." Of course the question texts are not the same, but nevertheless the answer is there. – rschwieb Jun 08 '22 at 15:20
1 Answers
3
There is no such a method!
But for some special primes $p$ We can find
1)the numbers $3,5,7$ are primitive roots modulo every prime number of the form $2^n+1$ for natural $n\gt1$.
Moreover, for these primes every non residue number modulo the prime number is primitive root!
2)$p,q$ are odd prime numbers such that, $p=4q+1$, then the number $2$ is a primitive root modulo $p$.
3)$p,q$ are odd prime numbers such that, $p=2q+1$, then every non residue modulo $p$ is a primitive root modulo $p$ except one of them.
And many others...
hamid kamali
- 3,201
k1.M
- 5,428