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arXiv · 1908.07290

Linear independence results for certain sums of reciprocals of Fibonacci and Lucas numbers

Daniel Duverney·Yuta Suzuki·Yohei Tachiya
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Abstract

The aim of this paper is to give linear independence results for the values of certain series. As an application, we derive arithmetical properties of the sums of reciprocals of Fibonacci and Lucas numbers associated with certain coprime sequences $\{n_\ell\}_{\ell\geq1}$. For example, the three numbers \[ 1,\qquad\sum_{p\text{:prime}}^{}\frac{1}{F_{p^2}},\qquad\sum_{p\text{:prime}}^{}\frac{1}{L_{p^2}} \] are linearly independent over $\mathbb{Q}(\sqrt{5})$, where $\{F_n\}$ and $\{L_n\}$ are the Fibonacci and Lucas numbers, respectively.

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Daniel Duverney, Yuta Suzuki, Yohei Tachiya. 2019-08-20. Linear independence results for certain sums of reciprocals of Fibonacci and Lucas numbers. https://arxiv.org/abs/1908.07290

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