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 iswhere 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 ∈ Z[X ] and d ∈ 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.

The Solvability of Polynomial Pell’s Equation

“…We choose so that (Murru, 2019). Thus, it is natural generalizing the study of the polynomial Pell's equation to higher degrees.…”