Abstract
The polynomial Pell equation is
where D is a given integer polynomial and the solutions P, Q must be integer polynomials. A classical paper of Nathanson (Proc Am Math Soc 86:89–92, 1976) solved it when \(D(x) = x^2 + d\). We show that the Rédei polynomials can be used in a very simple and direct way for providing these solutions. Moreover, this approach allows us to find all the integer polynomial solutions when \(D(x) = f^2(x) + d\), for any \(f \in {\mathbb {Z}}[X]\) and \(d \in {\mathbb {Z}}\), generalizing the result of Nathanson. We are also able to find solutions of some generalized polynomial Pell equations introducing an extension of Rédei polynomials to higher degrees.
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References
Barbeau, E.J.: Pell’s Equation. Springer, New York (2003)
Barbero, S., Cerruti, U., Murru, N.: Solving the Pell equation via Rédei rational functions. Fibonacci Q. 48, 348–357 (2010)
Bellini, E., Murru, N.: An efficient and secure RSA-like cryptosystem exploiting Rédei rational functions over conics. Finite Fields Appl. 39, 179–194 (2016)
Dubickas, A., Steuding, J.: The polynomial Pell equation. Elem. Math. 59, 133–143 (2004)
Gaunet, M.L.: Formes cubiques polynomiales. C. R. Acad. Sci. Paris 311, 491–494 (1990)
Griffin, J.C., Gunatillake, G.: Orthogonal-type poltnomials and Pell equations. Integral Transforms Spec. Funct. 24, 796–806 (2013)
Hazama, F.: Pell equations for polynomials. Indag. Math. 8, 387–397 (1997)
Lidl, R., Mullen, G.L., Turnwald, G.: Dickson Polynomials, Pitman Monographs and Surveys in Pure and Applied Math, vol. 65. Longman, London (1993)
Mc Laughlin, J.: Polynomial solutions to Pell’s equation and fundamental units in real quadratic fields. J. Lond. Math. Soc. 67, 16–28 (2003)
Mollin, R.A.: Polynomial solutions for Pell’s equation revisited. Indian J. Pure Appl. Math. 28, 429–438 (1997)
Nathanson, M.B.: Polynomial Pell’s equation. Proc. Am. Math. Soc. 86, 89–92 (1976)
Nobauer, R.: Cryptanalysis of the Rédei scheme. Contrib. Gen. Algebra 3, 255–264 (1984)
Pastor, A.V.: Generalized Chebyshev polynomials and the Pell–Abel equation. Fundam. Prikl. Mat. 7, 1123–1145 (2001)
Ramasamy, A.M.S.: Polynomial solutions for the Pell’s equation. Indian J. Pure Appl. Math. 25, 577–581 (1994)
Rédei, L.: Über eindeuting umkehrbare polynome in endlichen korpen. Acta Sci. Math. 11, 85–92 (1946)
Topuzoglu, A., Winterhof, A.: Topics in geometry, coding theory and cryptography. Algebra Appl. 6, 135–166 (2006)
Webb, W.A., Yokota, H.: Polynomial Pell’s equation. Proc. Am. Math. Soc. 131, 993–1006 (2003)
Webb, W.A., Yokota, H.: Polynomial Pell’s equation II. J. Number Theory 106, 128–141 (2004)
Yokota, H.: Solutions of polynomial Pell’s equation. J. Number Theory 130, 2003–2010 (2010)
Zapponi, L.: Parametric solutions of Pell equations. Proc. Rom. Number Theory Assoc. 1, 43–48 (2016)
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The author is really grateful to the anonymous referee for the carefully reading of the paper and for all the suggestions that improved the presentation of the paper.
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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, 693–703 (2020). https://doi.org/10.1007/s11139-019-00233-1
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DOI: https://doi.org/10.1007/s11139-019-00233-1