On the Diophantine equation $G_n(x)=G_m(P(x))$: Higher-order recurrences
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- by Clemens Fuchs, Attila Pethő and Robert F. Tichy PDF
- Trans. Amer. Math. Soc. 355 (2003), 4657-4681 Request permission
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 \[ G_{n+d}(x)=A_{d-1}(x)G_{n+d-1}(x)+\cdots +A_0(x)G_{n}(x), \quad \mbox {for} n\geq 0,\] 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 \[ G_{n}(x)=G_{m}(P(x))\] 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.References
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Additional Information
- Clemens Fuchs
- Affiliation: Institut für Mathematik, Technische Universität Graz, Steyrergasse 30, 8010 Graz, Austria
- MR Author ID: 705384
- ORCID: 0000-0002-0304-0775
- Email: clemens.fuchs@tugraz.at
- Attila Pethő
- Affiliation: Institute for Mathematics and Informatics, University of Debrecen, H-4010 Debrecen, PO Box 12, Hungary
- MR Author ID: 189083
- Email: pethoe@math.klte.hu
- Robert F. Tichy
- Affiliation: Institut für Mathematik, Technische Universität Graz, Steyrergasse 30, 8010 Graz, Austria
- MR Author ID: 172525
- Email: tichy@tugraz.at
- 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 - © Copyright 2003 American Mathematical Society
- 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
- MathSciNet review: 1990766
Dedicated: Dedicated to Wolfgang M. Schmidt on the occasion of his 70th birthday.