Inverse problem for Pell equation and real quadratic fields of the least type

TL;DR

This paper solves the inverse problem for Pell equations by characterizing fundamental discriminants through quadratic progressions, using approximation techniques and interval analysis, offering a new approach compared to previous methods.

Contribution

It introduces a novel, simpler method to solve the inverse Pell problem using approximation and interval analysis, differing from prior symmetric sequence and continued fraction approaches.

Findings

Fundamental discriminants are characterized by quadratic progressions.

Almost all square-free integers are minimal elements in these progressions.

The new approach simplifies the solution process for the inverse Pell problem.

Abstract

The purpose of this article is to give the solutions of the inverse problem for Pellian equations. For any rational number $0< a/b < 1$, the fundamental discriminants $D$ satisfying $(\lfloor \sqrt{D} \rfloor b + a)^2 - D b^2 = 4$ are given in terms of a quadratic progression. There were studies about this problem based on symmetric sequences $\{a_1,\...,a_{l-1}\}$ and periodic continued fractions $[a_0,\bar{a_1,\...,a_{l-1},a_l}]$, but in this article we solve the problem in a completely different way with simpler parameters. The result is obtained by measuring the quality of approximation of a rational number to $\sqrt{d}$ or $\frac{1+\sqrt{d}}{2}$, and by defining a short interval attached to each rational number. On this formulation we also show that for almost all square-free integer $d$, $d$ is the least element of the prescribed quadratic progression for some $a/b$.