On Integral Solutions of Ternary Quadratic Diophantine Equation

Abstract

In this paper, we analyze the ternary quadratic homogeneous Diophantine equation for finding its non – zero integer solutions. The proposed equation represents a homogeneous cone and its distinct integer solutions represent a distinct integer point on it. We obtain four different patterns of integer points on the cone. A few interesting properties of different patterns of solutions are presented with the help of special numbers such as  Polygonal number, Pyramidal number, Octahedral number, Pronic number, Stella Octangular number and Oblong number. Finally, the three triples of integers generated from the given solution are exhibited which satisfy the equation of the given cone.