LATTICE POINTS ON THE HOMOGENEOUS CUBIC EQUATION WITH FOUR UNKNOWNS푥푥2−푥푥푥푥+푥푥2+4푤푤2=8푧푧3

The Homogeneous cubic equation with four unknowns represented by the equation ????????2 − ???????????????? + ????????2 + 4????????2 = 8????????3 is analyzed for its patterns of non zero distinct integral solutions. Here we exhibit four different patterns. In each pattern we can find some interesting relations between the solutions and special numbers like Polygonal number, Three-Dimensional Figurate number, Star number, Rhombic Dodecahedral number etc.