Multi-variable Polynomial Solutions to Pell’s Equation and Fundamental Units in Real Quadratic Fields

Solving Pell’s equation is of relevance in finding fundamental units in real quadratic fields and for this reason polynomial solutions are of interest in that they can supply the fundamental units in infinite families of such fields. In this paper an algorithm is described which allows one to construct, for each positive integer n, a finite collection, {Fi}, of multi-variable polynomials (with integral coefficients), each satisfying a multi-variable polynomial Pell’s equation C i − FiH i = (−1)n−1, where Ci and Hi are multi-variable polynomials with integral coefficients. Each positive integer whose square-root has a regular continued fraction expansion with period n+1 lies in the range of one of these polynomials. Moreover, the continued fraction expansion of these polynomials is given explicitly as is the fundamental solution to the above multi-variable polynomial Pell’s equation. Some implications for determining the fundamental unit in a wide class of real quadratic fields is considered.