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 n, m, k with \(k \ge 3\).
Similar content being viewed by others
References
M.A. Alekseyev, On the intersection of Fibonacci. Pell and Lucas numbers, Integers 11(3), 239–259 (2011)
A. Baker, H. Davenport, The equations \(3x^2 - 2 = y^2\) and \(8x^2 - 7 = z^2\). Quart. J. Math. Oxford Ser. 2(20), 129–137 (1969)
E. Bravo, J.J. Bravo, F. Luca, Coincidences in generalized Lucas sequences. Fibonacci Quart. 52(4), 296–306 (2014)
J.J. Bravo, P. Das, S. Guzmán, S. Laishram, Powers in products of terms of Pell’s and Pell-Lucas Sequences. Int. J. Number Theory 11(4), 1259–1274 (2015)
J.J. Bravo, C.A. Gómez, F. Luca, Powers of two as sums of two \(k\)-Fibonacci numbers. Miskolc Math. Notes 17, 85–100 (2016)
J.J. Bravo, C.A. Gómez, J.L. Herrera, On the intersection of \(k\)-Fibonacci and Pell numbers. Bull. Korean Math. Soc. 56(2), 535–547 (2019)
J.J. Bravo, F. Luca, Powers of two in generalized Fibonacci sequences. Rev. Colombiana Mat. 46, 67–79 (2012)
J.J. Bravo, F. Luca, Repdigits in \(k\)-Lucas sequences. Proc. Indian Acad. Sci. Math. Sci. 124(2), 141–154 (2014)
A. Dujella, A. Pethő, A generalization of a theorem of Baker and Davenport. Quart. J. Math. Oxford Ser. (2) 49(195), 291–306 (1998)
F. Luca, V. Patel, On perfect powers that are sums of two Fibonacci numbers. J. Number Theory 189, 90–96 (2018)
E.M. Matveev, An explicit lower bound for a homogeneous rational linear form in the logarithms of algebraic numbers. II. Izv. Math. 64(6), 1217–1269 (2000)
E.P. Miles Jr., Generalized Fibonacci numbers and associated matrices. Amer. Math. Month. 67, 745–752 (1960)
M.D. Miller, Mathematical notes: On generalized Fibonacci numbers. Amer. Math. Month. 78, 1108–1109 (1971)
A. Pethő, Perfect powers in second order linear recurrences. J. Number Theory 15, 5–13 (1982)
S. E. Rihane, B. Faye, F. Luca and A. Togbé, Powers of two in generalized Lucas sequences, Fibonacci Quart 58(3), 254–260 (2020)
D.A. Wolfram, Solving generalized Fibonacci recurrences. Fibonacci Quart. 36(2), 129–145 (1998)
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
AT was supported in part by Purdue University Northwest.
Rights and permissions
About this article
Cite this article
Rihane, S.E., Togbé, A. On the intersection of k-Lucas sequences and some binary sequences. Period Math Hung 84, 125–145 (2022). https://doi.org/10.1007/s10998-021-00387-w
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10998-021-00387-w