Polynomial Pell’s equation
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- by William A. Webb and Hisashi Yokota PDF
- Proc. Amer. Math. Soc. 131 (2003), 993-1006 Request permission
Abstract:
Consider the polynomial Pell’s equation $X^2 -DY^2 = 1$, where $D = A^2 + 2C$ is a monic polynomial in ${\mathcal Z}[x]$ and $\deg {C} < \deg {A}$. Then for $A, C \in {\mathcal Q}[x]$, $\deg {C} < 2$, and $B = A/C \in {\mathcal Q}[x]$, a necessary and sufficient condition for the polynomial Pell’s equation to have a nontrivial solution in ${\mathcal Z}[x]$ is obtained.References
- N. H. Abel, Sur l’intégration de la formule différentielle $\rho dx/\sqrt {R}$, $R$ et $\rho$ étant des fonctions entières, in: Oeuvres Complètes de Niels Henrik Abel (L. Sylow and S. Lie, eds.). Christiania, t, 1 (1881), 104-144.
- Emil Artin, The collected papers of Emil Artin, Addison-Wesley Publishing Co., Inc., Reading, Mass.-London, 1965. Edited by Serge Lang and John T. Tate. MR 0176888
- Leonard E. Baum and Melvin M. Sweet, Continued fractions of algebraic power series in characteristic $2$, Ann. of Math. (2) 103 (1976), no. 3, 593–610. MR 409372, DOI 10.2307/1970953
- R. A. Mollin, Polynomial solutions for Pell’s equation revisited, Indian J. Pure Appl. Math. 28 (1997), no. 4, 429–438. MR 1448033
- Melvyn B. Nathanson, Polynomial Pell’s equations, Proc. Amer. Math. Soc. 56 (1976), 89–92. MR 401641, DOI 10.1090/S0002-9939-1976-0401641-4
- A. M. S. Ramasamy, Polynomial solutions for the Pell’s equation, Indian J. Pure Appl. Math. 25 (1994), no. 6, 577–581. MR 1285220
Additional Information
- William A. Webb
- Affiliation: Department of Mathematics, Washington State University, Pullman, Washington 99164
- Email: webb@math.wsu.edu
- Hisashi Yokota
- Affiliation: Department of Mathematics, Hiroshima Institute of Technology, 2-1-1 Miyake Saeki-ku Hiroshima, Japan
- Email: hyokota@cc.it-hiroshima.ac.jp
- Received by editor(s): April 3, 2001
- Published electronically: November 6, 2002
- Communicated by: David E. Rohrlich
- © Copyright 2002 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 131 (2003), 993-1006
- MSC (1991): Primary 11D25, 11A55
- DOI: https://doi.org/10.1090/S0002-9939-02-06934-4
- MathSciNet review: 1948087