POLYNOMIAL SOLUTIONS OF PELL'S EQUATION AND FUNDAMENTAL UNITS IN REAL QUADRATIC FIELDS

Abstract

Finding polynomial solutions of Pell's equation is of interest as such solutions sometimes allow the fundamental units to be determined in an infinite class of real quadratic fields.

In the paper, for each triple of positive integers $(c, h, f)$ satisfying $c^2-fh^2 = 1$ , where $(c, h)$ are the smallest pair of integers satisfying this equation, several sets of polynomials $(c(t), h(t), f(t))$ that satisfy $c(t)^2-f(t)h(t)^2 = 1$ and $(c(0), h(0), f(0)) = (c, h, f)$ are derived. Moreover, it is shown that the pair $(c(t), h(t))$ constitute the fundamental polynomial solution to the Pell equation above.

The continued fraction expansion of $\sqrt{f(t)}$ is given in certain general cases (for example when the continued fraction expansion of $\sqrt{f}$ has odd period length, or has even period length, or has period length $\equiv 2 \mod 4$ and the middle quotient has a particular form, etc.). Some applications to the determination of the fundamental unit in real quadratic fields is also discussed.