Journal of Integer Sequences, Vol. 12 (2009), Article 09.8.8

On the Equation axx (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


We study the solutions of the equation $ a^x\equiv x \left(\text{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

$\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 $ b$ to have analogous properties.

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(Concerned with sequences A133612 A133613 A133614 A133615 A133616 A133617 A133618 A133619 A144539 A144540 A144541 A144542 A144543 A144544 A151999 A152000.)

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

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