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

TL;DR

The paper identifies and analyzes LA2-type quadratic Diophantine equations, showing that, under the condition $A o ilde{u}^2 - au ilde{v}^2 = 1$ with $ au=D/4$ non-square, their integer solutions can be parameterized through Pell equation solutions. It proves that for $A o ext{Z}(1)$ the solution set $igr ext{D}_Aigl}$ is infinite and explicitly connected to Pell solutions $igl ext{P}_ auigr$, and derives an explicit counting formula for the number of lattice solutions in the rotated square region $|u|+|v|\le x$, valid for all sufficiently large $x$. The analysis decomposes $igr ext{D}_A(x)igr$ into five disjoint components, providing exact counts via logarithmic growth in the fundamental Pell parameters, and establishes an explicit threshold $oldsymbol{ extcal{L}}=igl\max_l M'_ligr)$ beyond which the counting formulas hold. These results yield a complete, computable description of integer solutions in large regions and reveal the structure of LA2-type equations via Pell-type dynamics.

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, 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 $\tilde{u} - τ\tilde{v}^2 = 1$. 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.

Theorems & Definitions (59)

• Remark 1.1
• Definition 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Remark 1.6
• proof
• Lemma 3.1
• proof
• Lemma 3.2