Coincidences in generalized Lucas sequences

TL;DR

This paper characterizes all integer coincidences between different generalized Lucas sequences, employing advanced number theory techniques to solve the associated Diophantine equations.

Contribution

It provides a complete classification of coincidences in generalized Lucas sequences, extending previous work with new bounds and methods.

Findings

All integer solutions to $L_n^{(k)}=L_m^{( ext{ell})}$ are explicitly determined.

The proof combines linear forms in logarithms with reduction techniques.

The results generalize known coincidences in classical Lucas sequences.

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].