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arXiv · 2001.10265

On the exponential Diophantine equation related to powers of two consecutive terms of Lucas sequences

Mahadi Ddamulira·Florian Luca
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Abstract

Let $r\ge 1$ be an integer and ${\bf U}:=(U_{n})_{n\ge 0} $ be the Lucas sequence given by $U_0=0$, $U_1=1, $ and $U_{n+2}=rU_{n+1}+U_n$, for all $ n\ge 0 $. In this paper, we show that there are no positive integers $r\ge 3,~x\ne 2,~n\ge 1$ such that $U_n^x+U_{n+1}^x$ is a member of ${\bf U}$.

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Mahadi Ddamulira, Florian Luca. 2020-04-23. On the exponential Diophantine equation related to powers of two consecutive terms of Lucas sequences. https://doi.org/10.1007/s11139-020-00278-7

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