Coincidences in generalized Lucas sequences

Abstract

For an integer $k\ge 2$, let $(L_n^{(k)}{)}_n$ be the $k-$generalized Lucas sequence which starts with $0,\dots ,0,2,1$ ($k$ 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 $k,\ell \ge 2$. 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].

arXiv comment14 pages