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 $X^{2}-DY^{2}=1$ found by the method of continued fractions for two squarefree polynomial expressions of radicands of Richaud-Degert type $D$ of the form $D=\left(f\left(u\right)\right)^{2}\pm2^{\alpha}n$, where $D$, $n>0$, $\alpha\geq0,\in\mathbb{Z}$, and $f\left(u\right)>0,\in\mathbb{Z}$, any polynomial function of $u\in\mathbb{Z}$ such that $f\left(u\right)\equiv0\left(mod\,\left(2^{\alpha-1}n\right)\right)$ or $f\left(u\right)\equiv\left(2^{\alpha-2}n\right)\left(mod\,\left(2^{\alpha-1}n\right)\right)$.