On the Equation ax ≡ x (mod bn)

We study the solutions of the equation $a^x\equiv x \left(\text{mod }b^{n}\right)$ . For some values of

, the solutions have a particularly rich structure. For example, for

we find that for every

that is not a multiple of

and for every $n\geq 2$ , the equation has just one solution

. Moreover, the solutions for different values of

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

we obtain

$\displaystyle 8\,^{56} \equiv 56 \left(\text{mod }10^2\right),\quad \qquad 8\,^... ... \quad\qquad 8\,^{5856} \equiv 5856 \left(\text{mod }10^{4}\right),\quad \dots$

In this paper we prove these results and provide sufficient conditions for the base

to have analogous properties.

Received June 10 2009; revised version received November 18 2009. Published in Journal of Integer Sequences, November 25 2009.

On the Equation a^x ≡ x (mod bⁿ)

J. Jiménez Urroz and J. Luis A. Yebra
Departament de Matemàtica Aplicada IV
Universitat Politècnica de Catalunya
Campus Nord, Edifici C3
C. Jordi Girona, 1-3
08034 Barcelona
Spain