Based upon choosing a prime $p$ of recommended length and $N=2p+1$, where $N$ is also prime, and $a^2$ is not congruent to $1 \pmod N$, and $a^p ≡ 1 \pmod N$, I believe you can rely upon $a$ as having order equal to $p$.
My question is, what is a good size for $a$? And the reason I ask is because I've heard that you want the base large enough so that most of the power possibilities exceed $N$. What I'm looking for is more formal requirements or an example that provides $p, N, a$ using current recommendations.