primitive roots of composite numbers

Question

I am looking for a way to find primitive roots of composite numbers by primitive roots of its prime factors.im looking for a analytic way no algebraic. I meant a way without meanings of abstract algebra like groups. it is a great help if you help me with your links or files.

Answer 1

The only numbers that admit a primitive root are those of the form $p^k$ and $2p^k$ with $p$ an odd prime, and $2$ and $4$.

Let's ignore the simple explicit cases $2$ and $4$.

Then we have the following paraphrased from the Wikipedia page:

If $g$ is a primitive root modulo $p$, then $g$ is a primitive root modulo all powers $p^k$, unless $g^{p – 1} \equiv 1 \pmod{p^2}$. In case $g^{p – 1} \equiv 1 \pmod{p^2}$ we have $g + p$ is primitive root.

If $g$ is a primitive root modulo $p^k$, then $g$ or $g + p^k$ (whichever one is odd) is a primitive root modulo $2p^k$.

Thus, the problem is easily reduced to the one for primes.