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.

多项式佩尔方程为

其中D是给定的整数多项式，解P，  Q必须是整数多项式。当\（D（x）= x ^ 2 + d \）时，Nathanson的经典论文（Proc Am Math Soc 86：89–92，1976）解决了该问题。我们证明Rédei多项式可以非常简单直接地用于提供这些解决方案。而且，对于任何\（f \ in {\ mathbb {Z}} [X] \中的任何\（D（x）= f ^ 2（x）+ d \），这种方法使我们能够找到所有整数多项式解。 ）和\（d \ in {\ mathbb {Z}} \）中，将Nathanson的结果推广。我们还能够找到将Rédei多项式扩展到更高程度的一些广义多项式Pell方程的解。