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Integer solutions of the generalized polynomial Pell equations and their finiteness: The quadratic case

Part of: Polynomials and matrices Diophantine equations Algebraic number theory: global fields

Published online by Cambridge University Press:  27 August 2025

Akanksha Gupta  and
Ekata Saha
Akanksha Gupta
Affiliation:
Department of Mathematics, https://ror.org/049tgcd06Indian Institute of Technology Delhi, Hauz Khas, New Delhi 110016, India e-mail: maz218514@maths.iitd.ac.in
Ekata Saha*
Affiliation:
Department of Mathematics, https://ror.org/049tgcd06Indian Institute of Technology Delhi, Hauz Khas, New Delhi 110016, India e-mail: maz218514@maths.iitd.ac.in
*
e-mail: ekata@maths.iitd.ac.in
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Abstract

Determining the polynomials $D \in {\mathbb Z}[x]$ such that the polynomial Pell equation ${P^2-DQ^2=1}$ has nontrivial solutions $P,Q$ in ${\mathbb Q}[x]$ (and in ${\mathbb Z}[x]$) is an open question. In this article, we consider the generalized polynomial Pell equation $P^2-DQ^2=n$, where $D \in {\mathbb Z}[x]$ is a monic quadratic polynomial and n is a nonzero integer. For $n=1$, such an equation always has nontrivial solutions in ${\mathbb Q}[x]$, but for a non-square integer n, the generalized polynomial Pell equation $P^2-DQ^2=n$ may not always have a solution in ${\mathbb Q}[x]$. Depending on n, we determine the polynomials $D=x^2+cx+d$, for which the equation $P^2-DQ^2=n$ has nontrivial solutions in ${\mathbb Q}[x]$ and in ${\mathbb Z}[x]$. Taking $n=-1$, this allows us to solve the negative polynomial Pell equation completely for any such D. An interesting feature is that there are certain polynomials D for which the generalized polynomial Pell equation has nontrivial solutions in ${\mathbb Z}[x]$, but only finitely many, whereas the solutions in ${\mathbb Q}[x]$ are infinitely many. Finally, we determine the monic quadratic polynomials D for which the solutions of $P^2-DQ^2=n$ in ${\mathbb Z}[x]$ exhibit this finiteness phenomenon.

Keywords

Polynomial Pell equationsPellian polynomialsinteger and rational solutions

MSC classification

Primary: 11C08: Polynomials 11D09: Quadratic and bilinear equations
Secondary: 11R09: Polynomials (irreducibility, etc.)

Information

Type
Article
Information
Canadian Mathematical Bulletin , First View , pp. 1 - 12
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Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of Canadian Mathematical Society

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References

Abel, N. H., Über die Integration der differential-formel $\rho dx/\sqrt{R}$ , wenn $R$ und $\rho$ ganze Funktionen sind . J. Reine Angew. Math. 1(1826), 185–221.Google Scholar
Ballini, F. and Veneziano, F., Hyperelliptic continued fractions in the singular case of genus zero . Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl. 33(2022), no. 4, 795–832.Google Scholar
Chebyshev, P. L., Sur l’intégration de la differential $\frac{x+A}{\sqrt{\left({x}^4+\alpha {x}^3+\beta {x}^2+\gamma \right.}} dx$ . J. Math. Pures Appl. 9(1864), no. 1, 225–246.Google Scholar
Dubickas, A. and Steuding, J., The polynomial Pell equation . Elem. Math. 59(2004), no. 4, 133–143.10.1007/s00017-004-0214-7CrossRefGoogle Scholar
Griffin, J. C. and Gunatillake, G., Orthogonal-type polynomials and Pell equations . Integral Transforms Spec. Funct. 24(2013), no. 10, 796–806.10.1080/10652469.2012.760083CrossRefGoogle Scholar
Hazama, F., Pell equations for polynomials . Indag. Math. (N.S.) 8(1997), no. 3, 387–397.10.1016/S0019-3577(97)81817-4CrossRefGoogle Scholar
Koymans, P. and Pagano, C., On Stevenhagen’s conjecture. Acta Mathematica (to appear).Google Scholar
McLaughlin, J., Polynomial solutions of Pell’s equation and fundamental units in real quadratic fields . J. London Math. Soc. (2) 67(2003), no. 1, 16–28.10.1112/S002461070200371XCrossRefGoogle Scholar
Mollin, R. A., Polynomial solutions for Pell’s equation revisited . Indian J. Pure Appl. Math. 28(1997), no. 4, 429–438.Google Scholar
Murru, N., A note on the use of Rédei polynomials for solving the polynomial Pell equation and its generalization to higher degrees . Ramanujan J. 53(2020), no. 3, 693–703.10.1007/s11139-019-00233-1CrossRefGoogle Scholar
Nathanson, M. B., Polynomial Pell’s equations . Proc. Amer. Math. Soc. 56(1976), 89–92.Google Scholar
Pastor, A. V., Generalized Chebyshev polynomials and the Pell-Abel equation . Fundam. Prikl. Mat. 7(2001), no. 4, 1123–1145.Google Scholar
Ramasamy, A. M. S., Polynomial solutions for the Pell’s equation . Indian J. Pure Appl. Math. 25(1994), no. 6, 577–581.Google Scholar
Schinzel, A., On some problems of the arithmetical theory of continued fractions II . Acta Arith. 7(1961/62), 287–298.10.4064/aa-7-3-287-298CrossRefGoogle Scholar
Scherr, Z. and Thompson, K., Quartic integral polynomial Pell equations . J. Number Theory 259(2024), 38–56.10.1016/j.jnt.2023.12.009CrossRefGoogle Scholar
Webb, W. A. and Yokota, H., Polynomial Pell’s equation . Proc. Amer. Math. Soc. 131(2003), no. 4, 993–1006.10.1090/S0002-9939-02-06934-4CrossRefGoogle Scholar
Webb, W. A. and Yokota, H., Polynomial Pell’s equation. II . J. Number Theory 106(2004), no. 1, 128–141.10.1016/j.jnt.2003.12.005CrossRefGoogle Scholar
Yokota, H., Solutions of polynomial Pell’s equation . J. Number Theory 130(2010), no. 9, 2003–2010.10.1016/j.jnt.2009.12.006CrossRefGoogle Scholar
Zannier, U., Hyperelliptic continued fractions and generalized Jacobians . Amer. J. Math. 141(2019), no. 1, 1–40.10.1353/ajm.2019.0000CrossRefGoogle Scholar
Zapponi, L., Parametric solutions of Pell equations . Proc. Roman Number Theory Assoc. 1(2016), 43–48.Google Scholar