On some general solutions of the simple Pell equation

Abstract

Two theorems are demonstrated giving analytical expressions of the fundamental solutions of the Pell equation $X^2-D{Y}^2=1$ found by the method of continued fractions for two squarefree polynomial expressions of radicands of Richaud-Degert type $D$ of the form $D={\left(f\left(u\right)\right)}^2\pm 2^{\alpha }n$, where $D$, $n>0$, $\alpha \ge 0,\in \mathbb{Z}$, and $f\left(u\right)>0,\in \mathbb{Z}$, any polynomial function of $u\in \mathbb{Z}$ such that $f\left(u\right)\equiv 0\left(mod\left(2^{\alpha -1}n\right)\right)$ or $f\left(u\right)\equiv \left(2^{\alpha -2}n\right)\left(mod\left(2^{\alpha -1}n\right)\right)$.

12 pages