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Czechoslovak Mathematical Journal, first online, pp. 1-16

$k$-Pell numbers of the form $x^a \pm x^b + 1$

Bibhu Prasad TRIPATHY, Bijan Kumar PATEL

Received June 12, 2025.   Published online September 30, 2026.
Abstract:  For an integer $k \geq2$, let $\{ P_n^{(k)} \}_n$ be the $k$-generalized Pell sequence which starts with $0, \dots,0,1$ ($k$ terms) and is given by the recurrence relation $P_n^{(k)} = 2 P_{n-1}^{(k)} +P_{n-2}^{(k)} +\dots+P_{n-k}^{(k)}$. We study the Diophantine equation $P_n^{(k)} = x^a \pm x^b + 1$ in positive integers $(n, k, x, a, b)$ with $k \geq3$, $0 \leq b < a$, and $2 \leq x \leq200$. This generalizes a result of B. Kafle, S. E. Rihane, A. Togbé (2020) regarding Pell numbers with the above property.
Keywords:  $k$-Pell number; linear form in logarithms; reduction method
Classification MSC:  11B39, 11J86
DOI:  10.21136/CMJ.2026.0219-25
PDF available at:  Institute of Mathematics CAS
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Affiliations:   Bibhu Prasad Tripathy, Department of Mathematics, School of Applied Sciences, KIIT University, Bhubaneswar, 751024, Odisha, India, e-mail: bptbibhu@gmail.com; Bijan Kumar Patel (corresponding author), P.G. Department of Mathematics, Government Women's College, Sundargarh, Sambalpur University, Odisha, India, e-mail: iiit.bijan@gmail.com
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