Abstract

We study the solutions of the equation a^{x} ≡ x (mod 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 ≥ 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≥0}, in the form x_{n}(a) = ∑_{i=0}^{n−1} x_{i}10^{i}. For instance, for a = 8 we obtain

8^{56} ≡ 56 (mod 10^{2}) ,  8^{856} ≡ 856 (mod 10^{3}) ,  8^{5856} ≡ 5856 (mod 10^{4}) , …

In this paper we prove these results and provide sufficient conditions for the base b to have analogous properties.