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arXiv · 1402.4085

Coincidences in generalized Lucas sequences

Eric F. Bravo·Jhon J. Bravo·Florian Luca
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Abstract

For an integer $k\geq 2$, let $(L_{n}^{(k)})_{n}$ be the $k-$generalized Lucas sequence which starts with $0,\ldots,0,2,1$ ($k$ terms) and each term afterwards is the sum of the $k$ preceding terms. In this paper, we find all the integers that appear in different generalized Lucas sequences; i.e., we study the Diophantine equation $L_n^{(k)}=L_m^{(\ell)}$ in nonnegative integers $n,k,m,\ell$ with $k, \ell\geq 2$. The proof of our main theorem uses lower bounds for linear forms in logarithms of algebraic numbers and a version of the Baker-Davenport reduction method. This paper is a continuation of the earlier work [4].

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Eric F. Bravo, Jhon J. Bravo, Florian Luca. 2014-02-17. Coincidences in generalized Lucas sequences. https://arxiv.org/abs/1402.4085

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