A note on the use of Rédei polynomials for solving the polynomial Pell equation and its generalization to higher degrees

Abstract

The polynomial Pell equation is \[P^2 - D Q^2 = 1\] where $D$ is a given integer polynomial and the solutions $P,Q$ must be integer polynomials. A classical paper of Nathanson \cite{Nat} 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 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.