Lattice Points on the Homogeneous Cone X2 Y2 40Z2

The ternary quadratic homogeneous equation representing homogeneous cone given by 2 2 2 X + Y = 40Z is analyzed for its nonzero distinct integer points on it . Seven different patterns of integer points satisfying the cone under consideration are obtained. A few interesting relation between the solutions and special number patterns namely Polygonal number , Pyramidal number , Octahedral number and Nasty number are presented. Also knowing an integer solution satisfying the given cone , two triples of integers generated from the given solution are exhibited.