Some Criteria for Class Numbers to Be Non-One

Abstract

Let $d$ be a positive integer which is not a perfect square and $n$ be any nonzero fixed integer. Then, the equation $x^2-\text{d}{y}^2=n$ is known as the general Pell equation. In this paper, we give some criteria for class numbers of certain real quadratic fields to be greater than one, depending on the solvability of the general Pell equation, ideals in quadratic orders, and the period length of the simple continued fraction expansions of $\sqrt{d}$ .