Solving the Pell equation via Rédei rational functions

Abstract: In this paper, we define a new product over $\mathbb{R}<sup>{\infty}$, which allows us to obtain a group isomorphic to $\mathbb R<sup>*$ with the usual product. This operation unexpectedly offers an interpretation of the R\'edei rational functions, making more clear some of their properties, and leads to another product, which generates a group structure over the Pell hyperbola. Finally, we join together these results, in order to evaluate solutions of Pell equation in an original way.