Show that there is no modulus with an odd prime number of primitive roots. I'm not really sure how to approach this at all. I was thinking of using the property that there exists a primitive root modulo $m$ if and only if $m = 2, 4$ or $m$ is of the form $p^k$ or $2p^k$, but I'm not really sure what to do next.
Show that there is no modulus with an odd prime number of primitive roots
modular arithmeticprimitive-roots
Best Answer
If there are any primitive roots modulo $m$ at all, then there are $\varphi(\varphi(m))$ of them. Being a totient, it could never be an odd number greater than $1$, regardless of primality.