What is a set of generators for $PSL(2,p)$ for $p$ prime?

Question

What is a set of generators for $PSL(2,p)$ when $p$ is prime, in particular when $p$ is a Mersenne prime? I know that for $p=7=2^3-1$, we can view $PSL(2,7)$ as the group of linear fractional transformations and the generators are $k\mapsto -1/k$, $k\mapsto k+1$, and $k\mapsto 2k$. This works because the quadratic residues modulo 7 are powers of 2, $\{1,2,4\}$. How does this generalize, for example to $p=31$?

Answer 1

Any $SL(2,p)$ is a quotient of $SL(2,\mathbb{Z})$ under obvious coefficient reduction. (While it seems "obvious", this is a theorem and not entirely trivial. It also holds for larger dimensions.)

This means that for any prime $p$ you can just take the two generators $k\mapsto -1/k$ and $k\mapsto k+1$ for $PSL(2,p)$, regardless of quadratic residues.