I am interested in the mean period of a quadratic congruential generator. Let $X_{n+1} = \sum_{i=0}^2 a_i X_n^i \bmod m$ where the $a_i \in \mathbb{Z_m}$ are chosen uniformly at random and $m$ is a fixed prime. Take $X_0$ to be uniformly selected from $\{0,\dots, m-1\}$.
Numerical simulations suggest the answer is asymptotically $\sqrt{\frac{\pi m}{2}}$ for large $m$. Is this a known result?
This result if provable is interesting for a number of reasons. One of these is that linear congruential generators have mean period length of $\Omega\left(\frac{n}{\log{\log{n}}}\right)$.
