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arXiv · 2401.06555

On a problem of Pillai involving S-units and Lucas numbers

Herbert Batte·Mahadi Ddamulira·Juma Kasozi·Florian Luca
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Abstract

Let $ \{L_n\}_{n\geq 0} $ be the sequence of Lucas numbers. In this paper, we look at the exponential Diophantine equation $L_n-2^x3^y=c$, for $n,x,y\in \mathbb{Z}_{\ge0}$. We treat the cases $c\in -\mathbb{N}$, $c=0$ and $c\in \mathbb{N}$ independently. In the cases that $c\in \mathbb{N}$ and $c\in -\mathbb{N}$, we find all integers $c$ such that the Diophantine equation has at least three solutions. These cases are treated independently since we employ quite different techniques in proving the two cases.

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Herbert Batte, Mahadi Ddamulira, Juma Kasozi, Florian Luca. 2024-01-12. On a problem of Pillai involving S-units and Lucas numbers. https://arxiv.org/abs/2401.06555

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