# Thread: find primitive roots for mod

1. ## find primitive roots for mod

is there an easy way to find primitive roots for a specific mod?

2. There isn't a specific formula to find primitive roots. However, say that you have found one primitive root, $\displaystyle g$ modulo $\displaystyle m$. Then, $\displaystyle g^k$ is also a primitive root iff $\displaystyle (\varphi (m), k) = 1$ and this gives the complete list.

More here: How to find a primitive root of N

3. So say I have found a primitive root mod 29. namely 2.

then the other primitive roots mod 29 are powers of 2?

that means that 8 is also a primitive root because 2^3=8 and (28,3)=1.

but 2^5=32 (28,5)=1 does that mean that 3 is a primitive root because 32 mod 29 is congruent to 3?

is the only other primitive root for 29 8?

4. Yes. $\displaystyle 2^k$ is a primitive root mod 29 iff $\displaystyle (k,28) = 1$.

So for example, $\displaystyle 2^3 \equiv 8$ is a primitive root since $\displaystyle (3, 28)=1$

5. but that is the only other p.r. besides 2. correct?

6. No ... there are many other values for $\displaystyle k$ such that $\displaystyle (k,28) = 1$.

Try $\displaystyle k = 1, 3, 5, 7, 9, 11, 13, ...$

In general, every prime $\displaystyle p$ has $\displaystyle \varphi (p-1)$ primitive roots.