Non-Trivial Integer Solutions of the Quadratic Diophantine Equations x2=25y2+29z2and2+2=612

Abstract

Many types of research have been devoted to finding the solutions η,ζ ,δ in the set of non-negative integers of Diophantine equations of the types η 2 = αζ 2 +βδ 2 and η 2 +ζ 2 = γδ 2 , where the fixed values η,ζ , γ,α,β and δ are integers. In this article, we will discuss integral solutions of the ternary quadratic Diophantine equation representing infinite cone given by x 2 = 25y 2 +29z 2 and lattice points of the ternary homogeneous Diophantine equation representing an infinite cone given by µ 2 +ϑ 2 = 61φ 2 are analyzed for its non-zero distinct lattice points. Few different patterns of integer points satisfying the infinite cone under consideration are obtained.