Generalized Pell's equations and Weber's class number problem

TL;DR

This paper generalizes Pell's equation using algebraic integers, introduces new continued fractions for these algebraic numbers, and connects the solutions to Weber's class number problem, providing explicit solutions and class number relations.

Contribution

It develops a new approach to generalized Pell's equations with algebraic integer coefficients and links the solutions to Weber's class number problem, offering explicit solutions and class number convergence results.

Findings

Explicit solutions to generalized Pell's equations over algebraic integers.

A new continued fraction expansion for algebraic numbers $X_n$.

A congruence relation for class number ratios in $ ext{Z}_2$-extensions.

Abstract

We study a generalization of Pell's equation, whose coefficients are certain algebraic integers. Let $X_0=0$ and $X_n=\sqrt{2+X_{n-1}}$ for each $n\in \mathbb{Z}_{\ge 1}$. We study the $\mathbb{Z}[X_{n-1}]$-solutions of the equation $x^2-X_n^2y^2=1$. By imitating the solution to the classical Pell's equation, we introduce new continued fraction expansions for $X_n$ over $\mathbb{Z}[X_{n-1}]$ and obtain an explicit solution of the generalized Pell's equation. In addition, we show that our explicit solution generates all the solutions if and only if the answer to Weber's class number problem is affirmative. We also obtain a congruence relation for the ratios of the class numbers of the $\mathbb{Z}_2$-extension over the rationals and show the convergence of the class numbers in $\mathbb{Z}_2$.