Geodesic
On the equation $a^x \equiv x\pmod{b^n}$
Journal of integer sequences, Tome 12 (2009) no. 8.

Voir la notice de l'article provenant de la source Electronic Library of Mathematics

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
Classification : 11A07
Keywords: exponential, congruences, integer sequences 
@article{JIS_2009__12_8_a6,
     author = {Jim\'enez Urroz, J. and Yebra, J.Luis A.},
     title = {On the equation $a^x \equiv x\pmod{b^n}$},
     journal = {Journal of integer sequences},
     publisher = {mathdoc},
     volume = {12},
     number = {8},
     year = {2009},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/JIS_2009__12_8_a6/}
}
TY  - JOUR
AU  - Jiménez Urroz, J.
AU  - Yebra, J.Luis A.
TI  - On the equation $a^x \equiv x\pmod{b^n}$
JO  - Journal of integer sequences
PY  - 2009
VL  - 12
IS  - 8
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/JIS_2009__12_8_a6/
LA  - en
ID  - JIS_2009__12_8_a6
ER  - 
%0 Journal Article
%A Jiménez Urroz, J.
%A Yebra, J.Luis A.
%T On the equation $a^x \equiv x\pmod{b^n}$
%J Journal of integer sequences
%D 2009
%V 12
%N 8
%I mathdoc
%U http://geodesic.mathdoc.fr/item/JIS_2009__12_8_a6/
%G en
%F JIS_2009__12_8_a6
Jiménez Urroz, J.; Yebra, J.Luis A. On the equation $a^x \equiv x\pmod{b^n}$. Journal of integer sequences, Tome 12 (2009) no. 8. http://geodesic.mathdoc.fr/item/JIS_2009__12_8_a6/