On the intersection of k-Lucas sequences and some binary sequences

Abstract

For an integer \(k\ge 2\), 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 k-generalized Lucas numbers which are Fibonacci, Pell or Pell–Lucas numbers, i.e., we study the Diophantine equations \(L_n^{(k)}=F_m\), \(L^{(k)}_n = P_m\) and \(L_n^{(k)}=Q_m\) in positive integers nmk with \(k \ge 3\).

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Correspondence to Salah Eddine Rihane.

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AT was supported in part by Purdue University Northwest.

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Rihane, S.E., Togbé, A. On the intersection of k-Lucas sequences and some binary sequences. Period Math Hung (2021). https://doi.org/10.1007/s10998-021-00387-w

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Keywords

  • k-generalized Lucas numbers
  • Linear form in logarithms
  • Reduction method

Mathematics Subject Classification

  • 11B39
  • 11J86