\documentclass[12pt,reqno]{article}

\usepackage{amsmath}

\def\ds{\displaystyle}
\def\beq{\begin{equation}}
\def\eeq{\end{equation}}
\def\tm#1{\left(\text{mod }#1\right)}

\begin{document}

\begin{abstract}
We study the solutions of the equation $a^x\equiv x \tm
{b^{n}}$. For some values of $b$, the solutions
have a particularly rich structure.  For example, for $b=10$ we find that for
every $a$ that is not a multiple of $10$ and for every $n\geq 2$, the
equation has just one solution $x_n(a)$. Moreover, the solutions for
different values of $n$ arise from a sequence $x(a)=
\{x_{i}\}_{i\geq 0}$, in the form $x_n(a)=\sum_{i=0}^{n-1} x_i 10^i$.
For instance, for $a=8$ we obtain $$ 8\,^{56}  \equiv 56 \tm
{10^2},\quad \qquad 8\,^{856}  \equiv 856 \tm {10^{3}}, \quad\qquad
8\,^{5856}  \equiv 5856 \tm {10^{4}},\quad \dots $$ 
In this paper we
prove these results and provide sufficient conditions for the base $b$
to have analogous properties.
\end{abstract}

\end{document}
