Abstract
We find all the solutions of the title Diophantine equation \(P_1^p+2P_2^p + \cdots +kP_k^p=P_n^q\) in positive integer variables \((k, n)\), where \(P_i\) is the \(i^{th}\) term of the Pell sequence if the exponents p, q are included in the set \(\{1,2\}\).
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Dasdemir, A.: On Pell, Pell-Lucas and modified Pell numbers by matrix methods. Appl. Math. Sci. 5, 3173–3181 (2011)
K. Gueth, F. Luca and L. Szalay, Solutions to \(F_1^p+2F_2^p + \cdots +kF_k^p=F_n^q\) with small given exponents (preprint)
Horadam, A.F.: Pell identities. Fibonacci Quart. 9, 245–263 (1971)
T. Koshy, Pell and Pell–Lucas Numbers with Applications, Springer-Verlag (New York, 2014)
I. Niven, H. S. Zuckerman and H. L. Montgomery, An Introduction to the Theory of Numbers, fifth edition, John Wiley & Sons (New York, 1991)
Soydan, G., Németh, L., Szalay, L.: On the Diophantine equation \(\sum _{j=1}^{k}jF_j^p=F_n^q\). Arch. Math. (Brno) 54, 177–188 (2018)
Acknowledgement
The authors thank the referee for the useful comments to improve the quality of this paper. The second author was supported by Purdue University Northwest.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Tchammou, E., Togbé, A. On the Diophantine equation \(\sum_{j=1}^{k}jP_j^p=P_n^q\) . Acta Math. Hungar. 162, 647–676 (2020). https://doi.org/10.1007/s10474-020-01043-4
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10474-020-01043-4