We study the solutions of the equation
![$ a^x\equiv x \left(\text{mod }b^{n}\right)$](img1.gif)
. For some values of
![$ b$](img2.gif)
, the solutions
have a particularly rich structure. For example, for
![$ b=10$](img3.gif)
we find that for
every
![$ a$](img4.gif)
that is not a multiple of
![$ 10$](img5.gif)
and for every
![$ n\geq 2$](img6.gif)
, the
equation has just one solution
![$ x_n(a)$](img7.gif)
. Moreover, the solutions for
different values of
![$ n$](img8.gif)
arise from a sequence
![$ x(a)=
\{x_{i}\}_{i\geq 0}$](img9.gif)
, in the form
![$ x_n(a)=\sum_{i=0}^{n-1} x_i 10^i$](img10.gif)
.
For instance, for
![$ a=8$](img11.gif)
we obtain
In this paper we
prove these results and provide sufficient conditions for the base
![$ b$](img2.gif)
to have analogous properties.