On Finding Integer Solutions to Homogeneous Ternary Quadratic Diophantine Equation

Abstract

The theoretical importance of polynomial equations of second degree in three unknowns with integral coefficients is great as they are closely connected with many problems of number theory. Specifically, the second degree polynomial equations with three unknowns in connection with geometrical figures occupy a pivotal role in the region of mathematics. The successful completion of exhibiting all integers satisfying the requirements set forth in the problem add to further progress of Number Theory as they offer good applications in the field of Graph theory, Modular theory, Coding and Cryptography, Engineering, Music and so on. Integers have repeatedly played a crucial role in the evolution of the Natural Sciences. The theory of integers provides answers to real world problems. Objectives: The objective of this research paper is to obtain varieties of integer solutions to homogeneous ternary quadratic Diophantine equation represented by the proposed equation . Geometrically, the considered polynomial equations of degree two with three unknowns represents cone. Methods: Various choices of integer solutions are secure from beginning to end employing linear modifications and used to simplify expressions. Patterns of solutions in integers are obtained by reducing the given polynomial equation to the equation which is solvable through employing suitable transformations and applying the factorization method. Findings: Six distinct transformations are applied to obtain choices of integral solutions for the considered second degree equation having three unknowns. Novelty: The equation in title has been reduced to either solvable ternary quadratic equation or a system of simultaneous equations through employing suitable transformations. Many lattice points satisfying the given cone are obtained through analytical process by means of substitution strategy and method of factorization.