  In 2009, Sol\'{e} and Zinoviev (\emph{Eur. J. Combin.}, vol. 30, no. 2, pp.
458-467, 2009) proposed an open problem of arithmetic interest to study the
period of the inversive pseudorandom number generators (IPRNGs) and to give
conditions bearing on $a, b$ to achieve maximal period, we focus on resolving
this open problem. In this paper, the period distribution of the IPRNGs over
the Galois ring $({\rm Z}_{p^{e}},+,\times)$ is considered, where $p>3$ is a
prime and $e\geq 2$ is an integer. The IPRNGs are transformed to 2-dimensional
linear feedback shift registers (LFSRs) so that the analysis of the period
distribution of the IPRNGs is transformed to the analysis of the period
distribution of the LFSRs. Then, by employing some analytical approaches, the
full information on the period distribution of the IPRNGs is obtained, which is
to make exact statistics about the period of the IPRNGs then count the number
of IPRNGs of a specific period when $a$, $b$ and $x_{0}$ traverse all elements
in ${\rm Z}_{p^{e}}$. The analysis process also indicates how to choose the
parameters and the initial values such that the IPRNGs fit specific periods.
