Integer roots of LA2-type function in the closed rotated square region

TL;DR

This paper investigates integer solutions of a specific class of Diophantine equations called LA2-type, especially those reducible to Pell's equation, and characterizes their solutions within certain geometric regions.

Contribution

It introduces properties of LA2-type equations, characterizes solutions for those reducible to Pell's equation, and provides a method to count solutions in bounded regions.

Findings

Solutions can be characterized for equations reducible to Pell's form.

A formula for counting solutions within |u| + |v| ≤ x is established.

The set of solutions in bounded regions is explicitly described.

Abstract

Let $\mathcal{A}$ be the set of all Diophantine equations of the form $au^2 + buv + cv^2 + du + ev + f = 0$, where $a,b,c,d,e,f \in \mathbb{Z}$ and $a > 0$. One way to solve the equation $A \in \mathcal{A}$ is by applying Lagrange's method which was introduced over 200 years ago. In this paper, we consider a self-defined Diophantine equation $A \in \mathcal{A}$, which we called the $LA2$-type equation, motivated by results of Teckan, \"Ozko\c{c}, Fenolahy, Ramanantsoa and Totohasina. We provide some properties of $LA2$-type equations, and determine the set of integer solutions of equation $A \in \mathcal{Z}(1)$, where $\mathcal{Z}(1)$ is the set of all $LA2$-type equations such that $A$ can be rewrite as Pell's equation $\tilde{u} - \tau\tilde{v}^2 = 1$. In addition, we show that there exist positive integers $M'_l$, $l = 1,2,3,4$ such that for any $x \in \mathbb{R}, x \geq \mathcal{L} := \max\{M'_l: l \in \{1,2,3,4\}\}$, the formula of the number of pairwise integer solutions to the equation $A \in \mathcal{Z}(1)$ in the region enclosed by the equation $|u| + |v| \leq x$ in the $uv$ plane, can be determined and proved. As a consequence, we characterize the set of integer solutions satisfying $A \in \mathcal{Z}(1)$ in the region enclosed by the equation $|u| + |v| \leq x$ in the $uv$ plane.