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On the Diophantine equation $G_n(x)=G_m(P(x))$: Higher-order recurrences


Authors: Clemens Fuchs, Attila Petho and Robert F. Tichy
Journal: Trans. Amer. Math. Soc. 355 (2003), 4657-4681
MSC (2000): Primary 11D45; Secondary 11D04, 11D61, 11B37
DOI: https://doi.org/10.1090/S0002-9947-03-03325-7
Published electronically: June 10, 2003
MathSciNet review: 1990766
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Abstract: Let $\mathbf{K}$ be a field of characteristic $0$ and let $(G_{n}(x))_{n=0}^{\infty}$ be a linear recurring sequence of degree $d$ in $\mathbf{K}[x]$ defined by the initial terms $G_0,\ldots,G_{d-1}\in\mathbf{K}[x]$ and by the difference equation

\begin{displaymath}G_{n+d}(x)=A_{d-1}(x)G_{n+d-1}(x)+\cdots+A_0(x)G_{n}(x), \quad \mbox{for} \,\, n\geq 0,\end{displaymath}

with $A_0,\ldots,A_{d-1}\in\mathbf{K}[x]$. Finally, let $P(x)$ be an element of $\mathbf{K}[x]$. In this paper we are giving fairly general conditions depending only on $G_0,\ldots,G_{d-1},$ on $P$, and on $A_0,\ldots,A_{d-1}$ under which the Diophantine equation

\begin{displaymath}G_{n}(x)=G_{m}(P(x))\end{displaymath}

has only finitely many solutions $(n,m)\in \mathbb{Z}^{2},n,m\geq 0$. Moreover, we are giving an upper bound for the number of solutions, which depends only on $d$. This paper is a continuation of the work of the authors on this equation in the case of second-order linear recurring sequences.


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Additional Information

Clemens Fuchs
Affiliation: Institut für Mathematik, Technische Universität Graz, Steyrergasse 30, 8010 Graz, Austria
Email: clemens.fuchs@tugraz.at

Attila Petho
Affiliation: Institute for Mathematics and Informatics, University of Debrecen, H-4010 Debrecen, PO Box 12, Hungary
Email: pethoe@math.klte.hu

Robert F. Tichy
Affiliation: Institut für Mathematik, Technische Universität Graz, Steyrergasse 30, 8010 Graz, Austria
Email: tichy@tugraz.at

DOI: https://doi.org/10.1090/S0002-9947-03-03325-7
Keywords: Diophantine equations, linear recurring sequences, $S$-unit equations
Received by editor(s): October 18, 2002
Received by editor(s) in revised form: February 7, 2003
Published electronically: June 10, 2003
Additional Notes: This work was supported by the Austrian Science Foundation FWF, grant S8307-MAT
The second author was supported by the Hungarian National Foundation for Scientific Research, Grant Nos. 29330 and 38225
Dedicated: Dedicated to Wolfgang M. Schmidt on the occasion of his 70th birthday.
Article copyright: © Copyright 2003 American Mathematical Society

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