Closed formulae for certain Fermat-Pell equations

TL;DR

This paper derives explicit formulas for certain Fermat-Pell equations by characterizing the continued fraction expansions of square roots, revealing conditions on parameters for solutions to exist.

Contribution

It provides new closed-form expressions for solutions to specific Fermat-Pell equations based on continued fraction properties.

Findings

Solutions exist when k is even or 3 does not divide j.

Explicit formulas for d and e are given.

The smallest solutions to the Fermat-Pell equations are identified.

Abstract

Given positive integers $j,k$, with $j\geq 2$, we show that there are positive integers $d,e$ such that $\sqrt{d}$ has continued fraction expansion $\sqrt{d}=[e,\overline{k,\dots,k,2e}]$, with period $j$, if and only if $k$ is even or $3\nmid j$, in which case we give closed formulae to find all such $d,e$ as well as the smallest solution in positive integers to the Fermat-Pell equation $X^2-dY^2=(-1)^j$.