On some general solutions of the simple Pell equation

Vladimir Pletser

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arXiv · 1501.06051

On some general solutions of the simple Pell equation

Vladimir Pletser

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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 $D=\left(f\left(u\right)\right)^{2}\pm2^αn$, where $D$, $n>0$, $α\geq0,\in\mathbb{Z}$, and $f\left(u\right)>0,\in\mathbb{Z}$, any polynomial function of $u\in\mathbb{Z}$ such that $f\left(u\right)\equiv0\left(mod\,\left(2^{α-1}n\right)\right)$ or $f\left(u\right)\equiv\left(2^{α-2}n\right)\left(mod\,\left(2^{α-1}n\right)\right)$.

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Vladimir Pletser. 2015-01-24. On some general solutions of the simple Pell equation. https://arxiv.org/abs/1501.06051

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