A sufficient condition for $x^p-x-a$ to be primitive

Question

Let $p$ ba a prime number and $F_p$ be the finite field with $p$ elements. Characterize the set of $a\in F_p$ such that $f=x^p-x-a$ is a primitive polynomial i.e. $x$ generates the multiplicative group of $F_p[x]/(f)$.

Answer 1

Further to my hint above, that

The polynomial $x^p − x − a \in \mathbb{F}_p[x]$ is primitive if and only if $a$ is primitive in $\mathbb{F}_p$.

I found the result of Cao to be of use, I think, from $2010$

On the Order of the Polynomial $x^p − x − a$