Families of Ternary Quadratic Diophantine Equations: Existence and Parametric Methods

Abstract

This study examines a class of non-homogeneous quadratic Diophantine equations in three variables. By employing suitable algebraic transformations and analytical techniques, various structural characteristics of their non-zero integer solutions are established. The investigation reveals important properties governing the existence and behavior of these solutions. Furthermore, the results uncover noteworthy connections between the obtained solutions and certain polygonal number patterns, thereby providing additional insight into their underlying number-theoretic structure.