Let $m$ be an integer. The set $\{r_1, r_2, \dots, r_s\}$ is a complete residue system module $m$ if
$r_i \not\equiv r_j (\text{mod } m)$ for all $i \neq j$ and
for all $n \in \mathbb{N}$, there is $r_i \in \{r_1, r_2, \dots, r_s\}$ such that $n \equiv r_i (\text{mod }m)$.
Given $p$ a prime number, the question I am trying to answer is whether it is possible to find a complete residue system composed only by prime numbers.
I know that if $p = 7$, this is possible. Indeed, $\{1, 2, 3, 5, 7, 11, 13\}$ is a complete residue system. I constructed this complete residue system from the "trivial" one, which is given by $\{0, 1, 2, 3, 4, 5, 6, 7\}$, then I disregarded the ones that are not prime, but I noticed that if I summed $7$ in each one of this, I would obtain a prime number. This seems to specific to me, so that I think I cannot do the same construction for a general prime number. On the other hand, I am not sure if this is the only way to do this.
Thank you in advance!