Summary: We study the solutions of the equation $ a^x\equiv x \left({mod }b^{n}\right)$. 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

11A07

exponential, congruences, integer sequences