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

Abstract

For each positive integer $n$ it is shown how to construct a finite collection of multivariable polynomials $\{F_i:=F_i(t,X_1,...,X_{\lfloor \frac{n+1}{2}\rfloor })\}$ such that each positive integer whose squareroot has a continued fraction expansion with period $n+1$ lies in the range of exactly one of these polynomials. Moreover, each of these polynomials satisfy a polynomial Pell's equation $C_i^2-F_i{H}_i^2=(-1{)}^{n-1}$ (where $C_i$ and $H_i$ are polynomials in the variables $t,X_1,...,X_{\lfloor \frac{n+1}{2}\rfloor }$) and the fundamental solution can be written down. Likewise, if all the $X_i$'s and $t$ are non-negative then the continued fraction expansion of $\sqrt{F_i}$ can be written down. Furthermore, the congruence class modulo 4 of $F_i$ depends in a simple way on the variables $t,X_1,...,X_{\lfloor \frac{n+1}{2}\rfloor }$ so that the fundamental unit can be written down for a large class of real quadratic fields. Along the way a complete solution is given to the problem of determining for which symmetric strings of positive integers $a_1,...,a_n$ do there exist positive integers $D$ and $a_0$ such that $\sqrt{D}=[a_0;\overline{a_1,>...,a_n,2a_0}]$.

13 pages

Cited by in corpus (2)

• Higher Descent on Pell Conics I. From Legendre to Selmer
• Higher Descent on Pell Conics II. Two Centuries of Missed Opportunities