Pell Equations and ℱ p l -Continued Fractions

Abstract

In this note, the solvability of the Pell equation, $X^2-D{Y}^2=1$ , is discussed over $\mathbb{Z}\times p^l\mathbb{Z}$ . In particular, we show that this equation is solvable over $\mathbb{Z}\times p^l\mathbb{Z}$ for each prime $p$ and natural number $l$ . Moreover, we show that solutions to the Pell equation over $\mathbb{Z}\times p^l\mathbb{Z}$ are completely determined by the ${\mathrm{ℱ}}_{p^l}$ -continued fraction expansion of $\sqrt{D}$ .