On the Diophantine equation 𝑈_{𝑛}-𝑏^{𝑚}=𝑐

Heintze, Sebastian; Tichy, Robert; Vukusic, Ingrid; Ziegler, Volker

doi:10.1090/mcom/3854

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.

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