# Existence of semi-primitive primes modulo a special class of numbers

Let $p$ be a prime and $N$ be an integer.

Then $p$ is called semi-primitive modulo $N$ if there exists a positive integer $j$ such that $p^j \equiv -1 \pmod{N}$.

Now let $m$ be a positive integer and let $N = \operatorname{rad}\left(\dfrac{p^m-1}{p-1}\right)$, where $\operatorname{rad}(c)$ denotes the product of all the (distinct) prime divisors of an integer $c$.

Question: What are the solutions $(p,m)$ to the equation: $$p^j \equiv -1 \pmod{\operatorname{rad}\left(\frac{p^m-1}{p-1}\right)}?$$ That is, for which $(p,m)$ is $p$ semi-primitive modulo $\operatorname{rad}\left(\frac{p^m-1}{p-1}\right)$?

One particular example is the case when $p$ is a Mersenne prime and $m = 4$.

The proof is a bit tedious but it turns out that this only works when $m = 2$ or when $p$ is a Mersenne prime and $m = 4$...