On some general solutions of the simple Pell equation

TL;DR

This paper derives analytical formulas for fundamental solutions of the Pell equation for specific polynomial radicands using continued fractions, expanding understanding of solutions for Richaud-Degert type D.

Contribution

It introduces new analytical expressions for Pell equation solutions with polynomial radicands of Richaud-Degert type, generalizing previous methods.

Findings

Provides explicit formulas for solutions with polynomial radicands.

Extends continued fraction methods to new classes of D.

Enhances understanding of Pell solutions for specific polynomial forms.

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 $f\left(u\right)\equiv\left(2^{\alpha-2}n\right)\left(mod\,\left(2^{\alpha-1}n\right)\right)$.