Some new results on the negative polynomial Pell’s equation

K. Anitha, I. Mumtaj Fathima and A. R. Vijayalakshmi
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 30, 2024, Number 2, Pages 223–235
DOI: 10.7546/nntdm.2024.30.2.223-235
Full paper (PDF, 259 Kb)


Authors and affiliations

K. Anitha
Department of Mathematics, SRM IST Ramapuram
Chennai 600089, India

I. Mumtaj Fathima
Research Scholar, Department of Mathematics,
Sri Venkateswara College of Engineering (Affiliated to Anna University)
Sriperumbudur, Chennai 602117, India

A. R. Vijayalakshmi
Department of Mathematics, Sri Venkateswara College of Engineering
Sriperumbudur, Chennai 602117, India


We consider the negative polynomial Pell’s equation P^2(X)-D(X)Q^2(X)=-1, where D(X)\in \mathbb{Z}[X] be some fixed, monic, square-free, even degree polynomials. In this paper, we investigate the existence of polynomial solutions P(X), Q(X) with integer coefficients.


  • Pell’s equation
  • Polynomial Pell’s equation
  • Gaussian integers
  • ABC conjecture for polynomials

2020 Mathematics Subject Classification

  • 11A99
  • 11C08
  • 11D99


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Manuscript history

  • Received: 29 October 2022
  • Revised: 28 March 2024
  • Accepted: 24 April 2024
  • Online First: 7 May 2024

Copyright information

Ⓒ 2024 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

Anitha, K., Mumtaj Fathima, I., & Vijayalakshmi, A. R. (2024). Some new results on the negative polynomial Pell’s equation. Notes on Number Theory and Discrete Mathematics, 30(2), 223-235, DOI: 10.7546/nntdm.2024.30.2.223-235.

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