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

Abstract

Let $\mathcal{A}$ be the set of all Diophantine equations of the form $a{u}^2+buv+c{v}^2+du+ev+f=0$, 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, Özkoç, 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 $\tilde{u}-\ddot{I}„{\tilde{v}}^2=1$. In addition, we show that there exist positive integers $M_l^{\prime }$, $l=1,2,3,4$ such that for any $x\in \mathbb{R},x\ge \mathcal{L}:=\max \{M_l^{\prime }:l\in \{1,2,3,4\}\}$, the formula of the number of pairwise integer solutions to the equation $A\in \mathcal{Z}(1)$ in the region enclosed by the equation $|u|+|v|\le x$ 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 $|u|+|v|\le x$ in the $uv$ plane.

45 pages