On the Diophantine equation $U_n - b^m = c$
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- by Sebastian Heintze, Robert F. Tichy, Ingrid Vukusic and Volker Ziegler
- Math. Comp. 92 (2023), 2825-2859
- DOI: https://doi.org/10.1090/mcom/3854
- Published electronically: May 15, 2023
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Abstract:
Let $(U_n)_{n\in \mathbb {N}}$ be a fixed linear recurrence sequence defined over the integers (with some technical restrictions). We prove that there exist effectively computable constants $B$ and $N_0$ such that for any $b,c\in \mathbb {Z}$ with $b> B$ the equation $U_n - b^m = c$ has at most two distinct solutions $(n,m)\in \mathbb {N}^2$ with $n\geq N_0$ and $m\geq 1$. Moreover, we apply our result to the special case of Tribonacci numbers given by $T_1= T_2=1$, $T_3=2$ and $T_{n}=T_{n-1}+T_{n-2}+T_{n-3}$ for $n\geq 4$. By means of the LLL-algorithm and continued fraction reduction we are able to prove $N_0=2$ and $B=e^{438}$. The corresponding reduction algorithm is implemented in Sage.References
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Bibliographic Information
- Sebastian Heintze
- Affiliation: Institute of Analysis and Number Theory, Graz University of Technology, Steyrergasse 30/II, A-8010 Graz, Austria
- MR Author ID: 1407704
- ORCID: 0000-0002-6356-1986
- Email: heintze@math.tugraz.at
- Robert F. Tichy
- Affiliation: Institute of Analysis and Number Theory, Graz University of Technology, Steyrergasse 30/II, A-8010 Graz, Austria
- MR Author ID: 172525
- Email: tichy@tugraz.at
- Ingrid Vukusic
- Affiliation: Department of Mathematics, University of Salzburg, Hellbrunnerstr. 34, A-5020 Salzburg, Austria
- MR Author ID: 1428653
- Email: ingrid.vukusic@plus.ac.at
- Volker Ziegler
- Affiliation: Department of Mathematics, University of Salzburg, Hellbrunnerstr. 34, A-5020 Salzburg, Austria
- MR Author ID: 744740
- ORCID: 0000-0002-6744-586X
- Email: volker.ziegler@plus.ac.at
- Received by editor(s): August 5, 2022
- Received by editor(s) in revised form: February 8, 2023, and March 23, 2023
- Published electronically: May 15, 2023
- Additional Notes: This work was supported by the Austrian Science Fund (FWF) under the project I4406.
- © Copyright 2023 American Mathematical Society
- Journal: Math. Comp. 92 (2023), 2825-2859
- MSC (2020): Primary 11Y50, 11D61, 11B37, 11J86
- DOI: https://doi.org/10.1090/mcom/3854