Diophantine equations for additive Pell numbers in Pell, Pell–Lucas, and Modified Pell numbers

Ahmet Emin and Ahmet Daşdemir
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 32, 2026, Number 1, Pages 112–119
DOI: 10.7546/nntdm.2026.32.1.112-119
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Authors and affiliations

Ahmet Emin
Department of Mathematics, Faculty of Engineering and Natural Sciences, Karabuk University
Karabük, Türkiye

Ahmet Daşdemir
Department of Mathematics, Faculty of Science, Kastamonu University
Kastamonu, Türkiye

Abstract

This paper investigates the Diophantine equations arising from ternary additive problems of Pell, Pell–Lucas, and Modified Pell numbers. Specifically, we characterize all integer solutions to the equation {P_n} + {P_m} + {P_r} = {X_k}, X \in \left\{ {{P},{Q},{R}} \right\}, where {P_i}, {Q_i}, and {R_i} denote the i-th terms of the Pell, Pell–Lucas, and Modified Pell sequences, respectively. By leveraging recurrence relations, Binet’s formulas, and Carmichael’s Primitive Divisor Theorem, we provide the first complete classification of solutions to this ternary additive problem. Our results reveal several parametric and singular solutions. Furthermore, we reduce prior results to binary sums of the form {P_n} + {P_m} = {X_k} as special instances of our framework.

Keywords

  • Pell number
  • Pell–Lucas number
  • Primitive divisor
  • Diophantine equation
  • Binet’s formula

2020 Mathematics Subject Classification

  • 11B39
  • 11D61

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Manuscript history

  • Received: 27 May 2025
  • Revised: 8 February 2026
  • Accepted: 26 February 2026
  • Online First: 26 February 2026

Copyright information

Ⓒ 2026 by the Authors.
This is an Open Access paper distributed under the terms and conditions of the Creative Commons Attribution 4.0 International License (CC BY 4.0).

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Cite this paper

Emin, A., & Daşdemir, A. (2026). Diophantine equations for additive Pell numbers in Pell, Pell–Lucas, and Modified Pell numbers. Notes on Number Theory and Discrete Mathematics, 32(1), 112-119, DOI: 10.7546/nntdm.2026.32.1.112-119.

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